1(* ::Package:: *) 2 3(* ::Title::Bold::Closed:: *) 4(*\[Integral]Sqrt[a Cos[c+d x]+b Sin[c+d x]]\[DifferentialD]x*) 5 6 7(* ::Subsubsection:: *) 8(*Derivation: Algebraic simplification*) 9 10 11(* ::Subsubsection:: *) 12(*Basis: a Cos[z]+b Sin[z]=Sqrt[a^2+b^2] Cos[z-ArcTan[a,b]]*) 13 14 15(* ::Subsubsection:: *) 16(*Rule: If a^2+b^2!=0 \[And] Sqrt[a^2+b^2]>0, then*) 17 18 19(* ::Subsubtitle::Bold:: *) 20(*\[Integral]Sqrt[a Cos[c+d x]+b Sin[c+d x]]\[DifferentialD]x \[LongRightArrow] (a^2+b^2)^(1/4)\[Integral]Sqrt[Cos[c+d x-ArcTan[a,b]]]\[DifferentialD]x*) 21 22 23(* ::Subsubsection:: *) 24(*Program code:*) 25 26 27(* ::Code:: *) 28Int[Sqrt[a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]],x_Symbol] := 29 Dist[(a^2+b^2)^(1/4),Int[Sqrt[Cos[c+d*x-ArcTan[a,b]]],x]] /; 30FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && PositiveQ[Sqrt[a^2+b^2]] 31 32 33(* ::Subsubsection:: *) 34(**) 35 36 37(* ::Subsubsection:: *) 38(*Derivation: Piecewise constant extraction and algebraic simplification*) 39 40 41(* ::Subsubsection:: *) 42(*Basis: \!\( *) 43(*\*SubscriptBox[\(\[PartialD]\), \(x\)]*) 44(*\*FractionBox[*) 45(*SqrtBox[\(a\ Cos[c + d\ x] + b\ Sin[c + d\ x]\)], *) 46(*SqrtBox[*) 47(*FractionBox[\(a\ Cos[c + d\ x] + b\ Sin[c + d\ x]\), *) 48(*SqrtBox[\( *) 49(*\*SuperscriptBox[\(a\), \(2\)] + *) 50(*\*SuperscriptBox[\(b\), \(2\)]\)]]]]\)=0*) 51 52 53(* ::Subsubsection:: *) 54(*Basis: (a Cos[z]+b Sin[z])/Sqrt[a^2+b^2]=Cos[z-ArcTan[a,b]]*) 55 56 57(* ::Subsubsection:: *) 58(*Rule: If a^2+b^2!=0 \[And] \[Not](Sqrt[a^2+b^2]>0), then*) 59 60 61(* ::Subsubtitle::Bold:: *) 62(*\[Integral]Sqrt[a Cos[c+d x]+b Sin[c+d x]]\[DifferentialD]x \[LongRightArrow] (Sqrt[a Cos[c+d x]+b Sin[c+d x]]/Sqrt[((a Cos[c+d x]+b Sin[c+d x])/Sqrt[a^2+b^2])])\[Integral]Sqrt[Cos[c+d x-ArcTan[a,b]]]\[DifferentialD]x*) 63 64 65(* ::Subsubsection:: *) 66(*Program code:*) 67 68 69(* ::Code:: *) 70(* Int[Sqrt[a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]],x_Symbol] := 71 Sqrt[a*Cos[c+d*x]+b*Sin[c+d*x]]/Sqrt[(a*Cos[c+d*x]+b*Sin[c+d*x])/Sqrt[a^2+b^2]]* 72 Int[Sqrt[Cos[c+d*x-ArcTan[a,b]]],x] /; 73FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && Not[PositiveQ[Sqrt[a^2+b^2]]] *) 74 75 76(* ::PageBreak:: *) 77(**) 78 79 80(* ::Title::Bold::Closed:: *) 81(*\[Integral]1/Sqrt[a Cos[c+d x]+b Sin[c+d x]] \[DifferentialD]x*) 82 83 84(* ::Subsubsection:: *) 85(*Derivation: Algebraic simplification*) 86 87 88(* ::Subsubsection:: *) 89(*Basis: a Cos[z]+b Sin[z]=Sqrt[a^2+b^2] Cos[z-ArcTan[a,b]]*) 90 91 92(* ::Subsubsection:: *) 93(*Rule: If a^2+b^2!=0 \[And] Sqrt[a^2+b^2]>0, then*) 94 95 96(* ::Subsubtitle::Bold:: *) 97(*\[Integral]1/Sqrt[a Cos[c+d x]+b Sin[c+d x]] \[DifferentialD]x \[LongRightArrow] (1/(a^2+b^2)^(1/4))\[Integral]1/Sqrt[Cos[c+d x-ArcTan[a,b]]] \[DifferentialD]x*) 98 99 100(* ::Subsubsection:: *) 101(*Program code:*) 102 103 104(* ::Code:: *) 105Int[1/Sqrt[a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]],x_Symbol] := 106 Dist[1/(a^2+b^2)^(1/4),Int[1/Sqrt[Cos[c+d*x-ArcTan[a,b]]],x]] /; 107FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && PositiveQ[Sqrt[a^2+b^2]] 108 109 110(* ::Subsubsection:: *) 111(**) 112 113 114(* ::Subsubsection:: *) 115(*Derivation: Piecewise constant extraction and algebraic simplification*) 116 117 118(* ::Subsubsection:: *) 119(*Basis: \!\( *) 120(*\*SubscriptBox[\(\[PartialD]\), \(x\)]*) 121(*\*FractionBox[*) 122(*SqrtBox[*) 123(*FractionBox[\(a\ Cos[c + d\ x] + b\ Sin[c + d\ x]\), *) 124(*SqrtBox[\( *) 125(*\*SuperscriptBox[\(a\), \(2\)] + *) 126(*\*SuperscriptBox[\(b\), \(2\)]\)]]], *) 127(*SqrtBox[\(a\ Cos[c + d\ x] + b\ Sin[c + d\ x]\)]]\)=0*) 128 129 130(* ::Subsubsection:: *) 131(*Basis: (a Cos[z]+b Sin[z])/Sqrt[a^2+b^2]=Cos[z-ArcTan[a,b]]*) 132 133 134(* ::Subsubsection:: *) 135(*Rule: If a^2+b^2!=0 \[And] \[Not](Sqrt[a^2+b^2]>0), then*) 136 137 138(* ::Subsubtitle::Bold:: *) 139(*\[Integral]1/Sqrt[a Cos[c+d x]+b Sin[c+d x]] \[DifferentialD]x \[LongRightArrow] (Sqrt[((a Cos[c+d x]+b Sin[c+d x])/Sqrt[a^2+b^2])]/Sqrt[a Cos[c+d x]+b Sin[c+d x]])\[Integral]1/Sqrt[Cos[c+d x-ArcTan[a,b]]] \[DifferentialD]x*) 140 141 142(* ::Subsubsection:: *) 143(*Program code:*) 144 145 146(* ::Code:: *) 147(* Int[1/Sqrt[a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]],x_Symbol] := 148 Sqrt[(a*Cos[c+d*x]+b*Sin[c+d*x])/Sqrt[a^2+b^2]]/Sqrt[a*Cos[c+d*x]+b*Sin[c+d*x]]* 149 Int[1/Sqrt[Cos[c+d*x-ArcTan[a,b]]],x] /; 150FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && Not[PositiveQ[Sqrt[a^2+b^2]]] *) 151 152 153(* ::PageBreak:: *) 154(**) 155 156 157(* ::Title::Bold::Closed:: *) 158(*\[Integral](a Cos[c+d x]+b Sin[c+d x])^n \[DifferentialD]x*) 159 160 161(* ::Subsubsection:: *) 162(*Rule: If a^2+b^2=0, then*) 163 164 165(* ::Subsubtitle::Bold:: *) 166(*\[Integral](a Cos[c+d x]+b Sin[c+d x])^n \[DifferentialD]x \[LongRightArrow] ((a (a Cos[c+d x]+b Sin[c+d x])^n)/(b d n))*) 167 168 169(* ::Subsubsection:: *) 170(*Program code:*) 171 172 173(* ::Code:: *) 174Int[(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_])^n_,x_Symbol] := 175 a*(a*Cos[c+d*x]+b*Sin[c+d*x])^n/(b*d*n) /; 176FreeQ[{a,b,c,d,n},x] && ZeroQ[a^2+b^2] 177 178 179(* ::Subsubsection:: *) 180(**) 181 182 183(* ::Subsubsection:: *) 184(*Reference: G&R 2.557.5b'*) 185 186 187(* ::Subsubsection:: *) 188(*Rule: If a^2+b^2!=0, then*) 189 190 191(* ::Subsubtitle::Bold:: *) 192(*\[Integral]1/(a Cos[c+d x]+b Sin[c+d x])^2 \[DifferentialD]x \[LongRightArrow] (Sin[c+d x]/(a d (a Cos[c+d x]+b Sin[c+d x])))*) 193 194 195(* ::Subsubsection:: *) 196(*Program code:*) 197 198 199(* ::Code:: *) 200Int[1/(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_])^2,x_Symbol] := 201 Sin[c+d*x]/(a*d*(a*Cos[c+d*x]+b*Sin[c+d*x])) /; 202FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] 203 204 205(* ::Subsubsection:: *) 206(**) 207 208 209(* ::Subsubsection:: *) 210(*Reference: G&R 2.557'*) 211 212 213(* ::Subsubsection:: *) 214(*Derivation: Integration by substitution*) 215 216 217(* ::Subsubsection:: *) 218(*Basis: If (n-1)/2\[Element]\[DoubleStruckCapitalZ], then (a Cos[z]+b Sin[z])^n=(a^2+b^2-(-b Cos[z]+a Sin[z])^2)^((n-1)/2) \!\( *) 219(*\*SubscriptBox[\(\[PartialD]\), \(z\)]\((\(-b\)\ Cos[z] + a\ Sin[z])\)\)*) 220 221 222(* ::Subsubsection:: *) 223(*Note: Should this rule also be used for odd n<0?*) 224 225 226(* ::Subsubsection:: *) 227(*Rule: If a^2+b^2!=0 \[And] (n-1)/2\[Element]\[DoubleStruckCapitalZ] \[And] n>0, then*) 228 229 230(* ::Subsubtitle::Bold:: *) 231(*\[Integral](a Cos[c+d x]+b Sin[c+d x])^n \[DifferentialD]x \[LongRightArrow] (1/d)Subst[Int[(a^2+b^2-x^2)^((n-1)/2),x],x,-b Cos[c+d x]+a Sin[c+d x]]*) 232 233 234(* ::Subsubsection:: *) 235(*Program code:*) 236 237 238(* ::Code:: *) 239Int[(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_])^n_,x_Symbol] := 240 Dist[1/d,Subst[Int[Regularize[(a^2+b^2-x^2)^((n-1)/2),x],x],x,-b*Cos[c+d*x]+a*Sin[c+d*x]]] /; 241FreeQ[{a,b},x] && NonzeroQ[a^2+b^2] && OddQ[n] && n>0 242 243 244(* ::Subsubsection:: *) 245(**) 246 247 248(* ::Subsubsection:: *) 249(*Derivation: Integration by parts with a double-back flip*) 250 251 252(* ::Subsubsection:: *) 253(*Rule: If a^2+b^2!=0 \[And] n>1 \[And] (n-1)/2\[NotElement]\[DoubleStruckCapitalZ], then*) 254 255 256(* ::Subsubtitle::Bold:: *) 257(*\[Integral](a Cos[c+d x]+b Sin[c+d x])^n \[DifferentialD]x \[LongRightArrow] *) 258(*-(((b Cos[c+d x]-a Sin[c+d x]) (a Cos[c+d x]+b Sin[c+d x])^(n-1))/(d n))+((n-1) (a^2+b^2))/n \[Integral](a Cos[c+d x]+b Sin[c+d x])^(n-2) \[DifferentialD]x*) 259 260 261(* ::Subsubsection:: *) 262(*Program code:*) 263 264 265(* ::Code:: *) 266Int[(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_])^n_,x_Symbol] := 267 -(b*Cos[c+d*x]-a*Sin[c+d*x])*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-1)/(d*n) + 268 Dist[(n-1)*(a^2+b^2)/n,Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n-2),x]] /; 269FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && RationalQ[n] && n>1 && Not[OddQ[n]] 270 271 272(* ::Subsubsection:: *) 273(**) 274 275 276(* ::Subsubsection:: *) 277(*Derivation: Integration by parts with a double-back flip*) 278 279 280(* ::Subsubsection:: *) 281(*Rule: If a^2+b^2!=0 \[And] n<-1 \[And] n!=-2, then*) 282 283 284(* ::Subsubtitle::Bold:: *) 285(*\[Integral](a Cos[c+d x]+b Sin[c+d x])^n \[DifferentialD]x \[LongRightArrow] *) 286(*(((b Cos[c+d x]-a Sin[c+d x]) (a Cos[c+d x]+b Sin[c+d x])^(n+1))/(d (n+1) (a^2+b^2)))+(n+2)/((n+1) (a^2+b^2)) \[Integral](a Cos[c+d x]+b Sin[c+d x])^(n+2) \[DifferentialD]x*) 287 288 289(* ::Subsubsection:: *) 290(*Program code:*) 291 292 293(* ::Code:: *) 294Int[(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_])^n_,x_Symbol] := 295 (b*Cos[c+d*x]-a*Sin[c+d*x])*(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+1)/(d*(n+1)*(a^2+b^2)) + 296 Dist[(n+2)/((n+1)*(a^2+b^2)),Int[(a*Cos[c+d*x]+b*Sin[c+d*x])^(n+2),x]] /; 297FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && RationalQ[n] && n<-1 && n!=-2 298 299 300(* ::PageBreak:: *) 301(**) 302 303 304(* ::Title::Bold::Closed:: *) 305(*\[Integral](Cos[c+d x]^m Sin[c+d x]^n)/(a Cos[c+d x]+b Sin[c+d x])^p \[DifferentialD]x*) 306 307 308(* ::Subsubsection:: *) 309(*Derivation: Algebraic expansion*) 310 311 312(* ::Subsubsection:: *) 313(*Basis: (Cos[z] Sin[z])/(a Cos[z]+b Sin[z])=(b Cos[z])/(a^2+b^2)+(a Sin[z])/(a^2+b^2)-(a b)/((a^2+b^2) (a Cos[z]+b Sin[z]))*) 314 315 316(* ::Subsubsection:: *) 317(*Rule: If a^2+b^2!=0 \[And] m,n\[Element]\[DoubleStruckCapitalZ] \[And] m>0 \[And] n>0, then*) 318 319 320(* ::Subsubtitle::Bold:: *) 321(*\[Integral](Cos[c+d x]^m Sin[c+d x]^n)/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x \[LongRightArrow] (b/(a^2+b^2))\[Integral]Cos[c+d x]^m Sin[c+d x]^(n-1) \[DifferentialD]x+ *) 322(* a/(a^2+b^2) \[Integral]Cos[c+d x]^(m-1) Sin[c+d x]^n \[DifferentialD]x-(a b)/(a^2+b^2) \[Integral](Cos[c+d x]^(m-1) Sin[c+d x]^(n-1))/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x*) 323 324 325(* ::Subsubsection:: *) 326(*Program code:*) 327 328 329(* ::Code:: *) 330(* Int[Cos[c_.+d_.*x_]^m_.*Sin[c_.+d_.*x_]^n_./(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]),x_Symbol] := 331 Dist[b/(a^2+b^2),Int[Cos[c+d*x]^m*Sin[c+d*x]^(n-1),x]] + 332 Dist[a/(a^2+b^2),Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^n,x]] - 333 Dist[a*b/(a^2+b^2),Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^(n-1)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x]] /; 334FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && IntegersQ[m,n] && m>0 && n>0 *) 335 336 337(* ::Subsubsection:: *) 338(**) 339 340 341(* ::Subsubsection:: *) 342(*Derivation: Algebraic expansion*) 343 344 345(* ::Subsubsection:: *) 346(*Basis: (Cos[z] Sin[z])/(a Cos[z]+b Sin[z])=(b Cos[z])/(a^2+b^2)+(a Sin[z])/(a^2+b^2)-(a b)/((a^2+b^2) (a Cos[z]+b Sin[z]))*) 347 348 349(* ::Subsubsection:: *) 350(*Rule: If a^2+b^2!=0 \[And] m,n,p\[Element]\[DoubleStruckCapitalZ] \[And] m>0 \[And] n>0 \[And] p<0, then*) 351 352 353(* ::Subsubtitle::Bold:: *) 354(*\[Integral]Cos[c+d x]^m Sin[c+d x]^n (a Cos[c+d x]+b Sin[c+d x])^p \[DifferentialD]x \[LongRightArrow] *) 355(*(b/(a^2+b^2))\[Integral]Cos[c+d x]^m Sin[c+d x]^(n-1) (a Cos[c+d x]+b Sin[c+d x])^(p+1) \[DifferentialD]x+*) 356(*a/(a^2+b^2) \[Integral]Cos[c+d x]^(m-1) Sin[c+d x]^n (a Cos[c+d x]+b Sin[c+d x])^(p+1) \[DifferentialD]x-*) 357(*(a b)/(a^2+b^2) \[Integral]Cos[c+d x]^(m-1) Sin[c+d x]^(n-1) (a Cos[c+d x]+b Sin[c+d x])^p \[DifferentialD]x *) 358 359 360(* ::Subsubsection:: *) 361(*Program code:*) 362 363 364(* ::Code:: *) 365Int[Cos[c_.+d_.*x_]^m_.*Sin[c_.+d_.*x_]^n_.*(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_])^p_,x_Symbol] := 366 Dist[b/(a^2+b^2),Int[Cos[c+d*x]^m*Sin[c+d*x]^(n-1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^(p+1),x]] + 367 Dist[a/(a^2+b^2),Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^n*(a*Cos[c+d*x]+b*Sin[c+d*x])^(p+1),x]] - 368 Dist[a*b/(a^2+b^2),Int[Cos[c+d*x]^(m-1)*Sin[c+d*x]^(n-1)*(a*Cos[c+d*x]+b*Sin[c+d*x])^p,x]] /; 369FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && IntegersQ[m,n,p] && m>0 && n>0 && p<0 370 371 372(* ::Subsubsection:: *) 373(**) 374 375 376(* ::Subsubsection:: *) 377(*Derivation: Algebraic expansion*) 378 379 380(* ::Subsubsection:: *) 381(*Basis: Sin[z]^2/(a Cos[z]+b Sin[z])=(b Sin[z])/(a^2+b^2)-(a Cos[z])/(a^2+b^2)+a^2/((a^2+b^2) (a Cos[z]+b Sin[z]))*) 382 383 384(* ::Subsubsection:: *) 385(*Rule: If a^2+b^2!=0 \[And] n\[Element]\[DoubleStruckCapitalZ] \[And] n>1, then*) 386 387 388(* ::Subsubtitle::Bold:: *) 389(*\[Integral](u Sin[c+d x]^n)/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x \[LongRightArrow] (b/(a^2+b^2))\[Integral]u Sin[c+d x]^(n-1) \[DifferentialD]x-*) 390(*a/(a^2+b^2) \[Integral]u Sin[c+d x]^(n-2) Cos[c+d x]\[DifferentialD]x+a^2/(a^2+b^2) \[Integral](u Sin[c+d x]^(n-2))/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x*) 391 392 393(* ::Subsubsection:: *) 394(*Program code:*) 395 396 397(* ::Code:: *) 398Int[u_.*Sin[c_.+d_.*x_]^n_./(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]),x_Symbol] := 399 Dist[b/(a^2+b^2),Int[u*Sin[c+d*x]^(n-1),x]] - 400 Dist[a/(a^2+b^2),Int[u*Sin[c+d*x]^(n-2)*Cos[c+d*x],x]] + 401 Dist[a^2/(a^2+b^2),Int[u*Sin[c+d*x]^(n-2)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x]] /; 402FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && IntegerQ[n] && n>0 && 403(n>1 || MatchQ[u,v_.*Tan[c+d*x]^m_. /; IntegerQ[m] && m>0]) 404 405 406(* ::Subsubsection:: *) 407(**) 408 409 410(* ::Subsubsection:: *) 411(*Derivation: Algebraic expansion*) 412 413 414(* ::Subsubsection:: *) 415(*Basis: Cos[z]^2/(a Cos[z]+b Sin[z])=(a Cos[z])/(a^2+b^2)-(b Sin[z])/(a^2+b^2)+b^2/((a^2+b^2) (a Cos[z]+b Sin[z]))*) 416 417 418(* ::Subsubsection:: *) 419(*Rule: If a^2+b^2!=0 \[And] n\[Element]\[DoubleStruckCapitalZ] \[And] n>1, then*) 420 421 422(* ::Subsubtitle::Bold:: *) 423(*\[Integral](u Cos[c+d x]^n)/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x \[LongRightArrow] (a/(a^2+b^2))\[Integral]u Cos[c+d x]^(n-1) \[DifferentialD]x-*) 424(*b/(a^2+b^2) \[Integral]u Cos[c+d x]^(n-2) Sin[c+d x]\[DifferentialD]x+b^2/(a^2+b^2) \[Integral](u Cos[c+d x]^(n-2))/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x*) 425 426 427(* ::Subsubsection:: *) 428(*Program code:*) 429 430 431(* ::Code:: *) 432Int[u_.*Cos[c_.+d_.*x_]^n_./(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]),x_Symbol] := 433 Dist[a/(a^2+b^2),Int[u*Cos[c+d*x]^(n-1),x]] - 434 Dist[b/(a^2+b^2),Int[u*Cos[c+d*x]^(n-2)*Sin[c+d*x],x]] + 435 Dist[b^2/(a^2+b^2),Int[u*Cos[c+d*x]^(n-2)/(a*Cos[c+d*x]+b*Sin[c+d*x]),x]] /; 436FreeQ[{a,b,c,d},x] && NonzeroQ[a^2+b^2] && IntegerQ[n] && n>0 && 437(n>1 || MatchQ[u,v_.*Cot[c+d*x]^m_. /; IntegerQ[m] && m>0]) 438 439 440(* ::Subsubsection:: *) 441(**) 442 443 444(* ::Subsubsection:: *) 445(*Derivation: Algebraic expansion*) 446 447 448(* ::Subsubsection:: *) 449(*Basis: Sec[z]/(a Cos[z]+b Sin[z])=Tan[z]/b+(b Cos[z]-a Sin[z])/(b (a Cos[z]+b Sin[z]))*) 450 451 452(* ::Subsubsection:: *) 453(*Rule:*) 454 455 456(* ::Subsubtitle::Bold:: *) 457(*\[Integral](u Sec[c+d x])/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x \[LongRightArrow] (1/b)\[Integral]u Tan[c+d x]\[DifferentialD]x+1/b \[Integral](u (b Cos[c+d x]-a Sin[c+d x]))/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x*) 458 459 460(* ::Subsubsection:: *) 461(*Program code:*) 462 463 464(* ::Code:: *) 465(* Int[u_.*Sec[c_.+d_.*x_]/(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]),x_Symbol] := 466 Dist[1/b,Int[u*Tan[c+d*x],x]] + 467 Dist[1/b,Int[u*(b*Cos[c+d*x]-a*Sin[c+d*x])/(a*Cos[c+d*x]+b*Sin[c+d*x]),x]] /; 468FreeQ[{a,b,c,d},x] *) 469 470 471(* ::Subsubsection:: *) 472(**) 473 474 475(* ::Subsubsection:: *) 476(*Derivation: Algebraic expansion*) 477 478 479(* ::Subsubsection:: *) 480(*Basis: Csc[z]/(a Cos[z]+b Sin[z])=Cot[z]/a-(b Cos[z]-a Sin[z])/(a (a Cos[z]+b Sin[z]))*) 481 482 483(* ::Subsubsection:: *) 484(*Rule:*) 485 486 487(* ::Subsubtitle::Bold:: *) 488(*\[Integral](u Csc[c+d x])/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x \[LongRightArrow] (1/a)\[Integral]u Cot[c+d x]\[DifferentialD]x-1/a \[Integral](u (b Cos[c+d x]-a Sin[c+d x]))/(a Cos[c+d x]+b Sin[c+d x]) \[DifferentialD]x*) 489 490 491(* ::Subsubsection:: *) 492(*Program code:*) 493 494 495(* ::Code:: *) 496(* Int[u_.*Csc[c_.+d_.*x_]/(a_.*Cos[c_.+d_.*x_]+b_.*Sin[c_.+d_.*x_]),x_Symbol] := 497 Dist[1/a,Int[u*Cot[c+d*x],x]] - 498 Dist[1/a,Int[u*(b*Cos[c+d*x]-a*Sin[c+d*x])/(a*Cos[c+d*x]+b*Sin[c+d*x]),x]] /; 499FreeQ[{a,b,c,d},x] *) 500 501 502(* ::PageBreak:: *) 503(**) 504 505 506(* ::Title::Bold::Closed:: *) 507(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x*) 508 509 510(* ::Subsubsection:: *) 511(*Reference: G&R 2.558.4c*) 512 513 514(* ::Subsubsection:: *) 515(*Rule: If a-b=0, then*) 516 517 518(* ::Subsubtitle::Bold:: *) 519(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] (1/(c e))Log[a+c Tan[1/2 (d+e x)]]*) 520 521 522(* ::Subsubsection:: *) 523(*Program code:*) 524 525 526(* ::Code:: *) 527Int[1/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 528 Log[a+c*Tan[(d+e*x)/2]]/(c*e) /; 529FreeQ[{a,b,c,d,e},x] && ZeroQ[a-b] 530 531 532(* ::Subsubsection:: *) 533(**) 534 535 536(* ::Subsubsection:: *) 537(*Reference: G&R 2.558.4c*) 538 539 540(* ::Subsubsection:: *) 541(*Rule: If a+b=0, then*) 542 543 544(* ::Subsubtitle::Bold:: *) 545(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] -(1/(c e))Log[a+c Cot[1/2 (d+e x)]]*) 546 547 548(* ::Subsubsection:: *) 549(*Program code:*) 550 551 552(* ::Code:: *) 553Int[1/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 554 -Log[a+c*Cot[(d+e*x)/2]]/(c*e) /; 555FreeQ[{a,b,c,d,e},x] && ZeroQ[a+b] 556 557 558(* ::Subsubsection:: *) 559(**) 560 561 562(* ::Subsubsection:: *) 563(*Reference: G&R 2.558.4d*) 564 565 566(* ::Subsubsection:: *) 567(*Rule: If a^2-b^2-c^2=0, then*) 568 569 570(* ::Subsubtitle::Bold:: *) 571(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] ((-c+a Sin[d+e x])/(c e (c Cos[d+e x]-b Sin[d+e x])))*) 572 573 574(* ::Subsubsection:: *) 575(*Program code:*) 576 577 578(* ::Code:: *) 579Int[1/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 580 (-c+a*Sin[d+e*x])/(c*e*(c*Cos[d+e*x]-b*Sin[d+e*x])) /; 581FreeQ[{a,b,c,d,e},x] && ZeroQ[a^2-b^2-c^2] 582 583 584(* ::Subsubsection:: *) 585(**) 586 587 588(* ::Subsubsection:: *) 589(*Reference: G&R 2.558.4a, CRC 342b*) 590 591 592(* ::Subsubsection:: *) 593(*Rule: If a^2-b^2!=0 \[And] a^2-b^2-c^2>0, then*) 594 595 596(* ::Subsubtitle::Bold:: *) 597(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] (2/(e Sqrt[a^2-b^2-c^2]))ArcTan[(c+(a-b) Tan[1/2 (d+e x)])/Sqrt[a^2-b^2-c^2]]*) 598 599 600(* ::Subsubsection:: *) 601(*Program code:*) 602 603 604(* ::Code:: *) 605Int[1/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 606 2*ArcTan[(c+(a-b)*Tan[(d+e*x)/2])/Rt[a^2-b^2-c^2,2]]/(e*Rt[a^2-b^2-c^2,2]) /; 607FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2] && PosQ[a^2-b^2-c^2] 608 609 610(* ::Subsubsection:: *) 611(**) 612 613 614(* ::Subsubsection:: *) 615(*Reference: G&R 2.558.4b', CRC 342b'*) 616 617 618(* ::Subsubsection:: *) 619(*Rule: If a^2-b^2!=0 \[And] \[Not](a^2-b^2-c^2>0), then*) 620 621 622(* ::Subsubtitle::Bold:: *) 623(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] -(2/(e Sqrt[-a^2+b^2+c^2]))ArcTanh[(c+(a-b) Tan[1/2 (d+e x)])/Sqrt[-a^2+b^2+c^2]]*) 624 625 626(* ::Subsubsection:: *) 627(*Program code:*) 628 629 630(* ::Code:: *) 631Int[1/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 632 -2*ArcTanh[(c+(a-b)*Tan[(d+e*x)/2])/Rt[-a^2+b^2+c^2,2]]/(e*Rt[-a^2+b^2+c^2,2]) /; 633FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2] && NegQ[a^2-b^2-c^2] 634 635 636(* ::PageBreak:: *) 637(**) 638 639 640(* ::Title::Bold::Closed:: *) 641(*\[Integral]Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]\[DifferentialD]x*) 642 643 644(* ::Subsubsection:: *) 645(*Reference: G&R 2.558.1 inverted with n=1/2 and a^2-b^2-c^2=0*) 646 647 648(* ::Subsubsection:: *) 649(*Rule: If a^2-b^2-c^2=0, then*) 650 651 652(* ::Subsubtitle::Bold:: *) 653(*\[Integral]Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]\[DifferentialD]x \[LongRightArrow] ((2 (-c Cos[d+e x]+b Sin[d+e x]))/(e Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]))*) 654 655 656(* ::Subsubsection:: *) 657(*Program code:*) 658 659 660(* ::Code:: *) 661Int[Sqrt[a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]],x_Symbol] := 662 2*(-c*Cos[d+e*x]+b*Sin[d+e*x])/(e*Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]) /; 663FreeQ[{a,b,c,d,e},x] && ZeroQ[a^2-b^2-c^2] 664 665 666(* ::Subsubsection:: *) 667(**) 668 669 670(* ::Subsubsection:: *) 671(*Derivation: Algebraic simplification*) 672 673 674(* ::Subsubsection:: *) 675(*Basis: a+b Cos[z]+c Sin[z]=a+Sqrt[b^2+c^2] Cos[z-ArcTan[b,c]]*) 676 677 678(* ::Subsubsection:: *) 679(*Rule: If a^2-b^2-c^2!=0 \[And] a+Sqrt[b^2+c^2]>0, then*) 680 681 682(* ::Subsubtitle::Bold:: *) 683(*\[Integral]Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]\[DifferentialD]x \[LongRightArrow] \[Integral]Sqrt[a+Sqrt[b^2+c^2] Cos[d+e x-ArcTan[b,c]]]\[DifferentialD]x*) 684 685 686(* ::Subsubsection:: *) 687(*Program code:*) 688 689 690(* ::Code:: *) 691Int[Sqrt[a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]],x_Symbol] := 692 Int[Sqrt[a+Sqrt[b^2+c^2]*Cos[d+e*x-ArcTan[b,c]]],x] /; 693FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2-c^2] && PositiveQ[a+Sqrt[b^2+c^2]] 694 695 696(* ::Subsubsection:: *) 697(**) 698 699 700(* ::Subsubsection:: *) 701(*Derivation: Piecewise constant extraction and algebraic simplification*) 702 703 704(* ::Subsubsection:: *) 705(*Basis: \!\( *) 706(*\*SubscriptBox[\(\[PartialD]\), \(x\)]*) 707(*\*FractionBox[*) 708(*SqrtBox[\(a + b\ Cos[d + e\ x] + c\ Sin[d + e\ x]\)], *) 709(*SqrtBox[*) 710(*FractionBox[\(a + b\ Cos[d + e\ x] + c\ Sin[d + e\ x]\), \(a + *) 711(*\*SqrtBox[\( *) 712(*\*SuperscriptBox[\(b\), \(2\)] + *) 713(*\*SuperscriptBox[\(c\), \(2\)]\)]\)]]]\)=0*) 714 715 716(* ::Subsubsection:: *) 717(*Basis: a+b Cos[z]+c Sin[z]=a+Sqrt[b^2+c^2] Cos[z-ArcTan[b,c]]*) 718 719 720(* ::Subsubsection:: *) 721(*Rule: If a^2-b^2-c^2!=0 \[And] \[Not](a+Sqrt[b^2+c^2]>0), then*) 722 723 724(* ::Subsubtitle::Bold:: *) 725(*\[Integral]Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]\[DifferentialD]x \[LongRightArrow] (Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]/Sqrt[((a+b Cos[d+e x]+c Sin[d+e x])/(a+Sqrt[b^2+c^2]))])\[Integral]Sqrt[a/(a+Sqrt[b^2+c^2])+Sqrt[b^2+c^2]/(a+Sqrt[b^2+c^2]) Cos[d+e x-ArcTan[b,c]]]\[DifferentialD]x*) 726 727 728(* ::Subsubsection:: *) 729(*Program code:*) 730 731 732(* ::Code:: *) 733Int[Sqrt[a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]],x_Symbol] := 734 Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/Sqrt[(a+b*Cos[d+e*x]+c*Sin[d+e*x])/(a+Sqrt[b^2+c^2])]* 735 Int[Sqrt[a/(a+Sqrt[b^2+c^2])+Sqrt[b^2+c^2]/(a+Sqrt[b^2+c^2])*Cos[d+e*x-ArcTan[b,c]]],x] /; 736FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2-c^2] && Not[PositiveQ[a+Sqrt[b^2+c^2]]] 737 738 739(* ::PageBreak:: *) 740(**) 741 742 743(* ::Title::Bold::Closed:: *) 744(*\[Integral]1/Sqrt[a+b Cos[d+e x]+c Sin[d+e x]] \[DifferentialD]x*) 745 746 747(* ::Subsubsection:: *) 748(*Derivation: Algebraic simplification NonzeroQ[a^2 - b^2 - c^2] ???? *) 749 750 751(* ::Subsubsection:: *) 752(*Basis: a+b Cos[z]+c Sin[z]=a+Sqrt[b^2+c^2] Cos[z-ArcTan[b,c]]*) 753 754 755(* ::Subsubsection:: *) 756(*Rule: If a+Sqrt[b^2+c^2]>0, then*) 757 758 759(* ::Subsubtitle::Bold:: *) 760(*\[Integral]1/Sqrt[a+b Cos[d+e x]+c Sin[d+e x]] \[DifferentialD]x \[LongRightArrow] \[Integral]1/Sqrt[a+Sqrt[b^2+c^2] Cos[d+e x-ArcTan[b,c]]] \[DifferentialD]x*) 761 762 763(* ::Subsubsection:: *) 764(*Program code:*) 765 766 767(* ::Code:: *) 768Int[1/Sqrt[a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]],x_Symbol] := 769 Int[1/Sqrt[a+Sqrt[b^2+c^2]*Cos[d+e*x-ArcTan[b,c]]],x] /; 770FreeQ[{a,b,c,d,e},x] && PositiveQ[a+Sqrt[b^2+c^2]] 771 772 773(* ::Subsubsection:: *) 774(**) 775 776 777(* ::Subsubsection:: *) 778(*Derivation: Piecewise constant extraction and algebraic simplification*) 779 780 781(* ::Subsubsection:: *) 782(*Basis: \!\( *) 783(*\*SubscriptBox[\(\[PartialD]\), \(x\)]*) 784(*\*FractionBox[*) 785(*SqrtBox[*) 786(*FractionBox[\(a + b\ Cos[d + e\ x] + c\ Sin[d + e\ x]\), \(a + *) 787(*\*SqrtBox[\( *) 788(*\*SuperscriptBox[\(b\), \(2\)] + *) 789(*\*SuperscriptBox[\(c\), \(2\)]\)]\)]], *) 790(*SqrtBox[\(a + b\ Cos[d + e\ x] + c\ Sin[d + e\ x]\)]]\)=0*) 791 792 793(* ::Subsubsection:: *) 794(*Basis: a+b Cos[z]+c Sin[z]=a+Sqrt[b^2+c^2] Cos[z-ArcTan[b,c]]*) 795 796 797(* ::Subsubsection:: *) 798(*Rule: If a+Sqrt[b^2+c^2]!=0 \[And] \[Not](a+Sqrt[b^2+c^2]>0), then*) 799 800 801(* ::Subsubtitle::Bold:: *) 802(*\[Integral]1/Sqrt[a+b Cos[d+e x]+c Sin[d+e x]] \[DifferentialD]x \[LongRightArrow] (Sqrt[((a+b Cos[d+e x]+c Sin[d+e x])/(a+Sqrt[b^2+c^2]))]/Sqrt[a+b Cos[d+e x]+c Sin[d+e x]])\[Integral]1/Sqrt[a/(a+Sqrt[b^2+c^2])+Sqrt[b^2+c^2]/(a+Sqrt[b^2+c^2]) Cos[d+e x-ArcTan[b,c]]] \[DifferentialD]x*) 803 804 805(* ::Subsubsection:: *) 806(*Program code:*) 807 808 809(* ::Code:: *) 810Int[1/Sqrt[a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]],x_Symbol] := 811 Sqrt[(a+b*Cos[d+e*x]+c*Sin[d+e*x])/(a+Sqrt[b^2+c^2])]/Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]* 812 Int[1/Sqrt[a/(a+Sqrt[b^2+c^2])+Sqrt[b^2+c^2]/(a+Sqrt[b^2+c^2])*Cos[d+e*x-ArcTan[b,c]]],x] /; 813FreeQ[{a,b,c,d,e},x] && NonzeroQ[a+Sqrt[b^2+c^2]] && Not[PositiveQ[a+Sqrt[b^2+c^2]]] 814 815 816(* ::PageBreak:: *) 817(**) 818 819 820(* ::Title::Bold::Closed:: *) 821(*\[Integral](a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x*) 822 823 824(* ::Subsubsection:: *) 825(*Reference: G&R 2.558.1 inverted with a^2-b^2-c^2=0*) 826 827 828(* ::Subsubsection:: *) 829(*Rule: If a^2-b^2-c^2=0 \[And] n\[Element]\[DoubleStruckCapitalF] \[And] n>1, then*) 830 831 832(* ::Subsubtitle::Bold:: *) 833(*\[Integral](a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x \[LongRightArrow] *) 834(* (((-c Cos[d+e x]+b Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^(n-1))/(e n))+*) 835(*(a (2 n-1))/n \[Integral](a+b Cos[d+e x]+c Sin[d+e x])^(n-1) \[DifferentialD]x *) 836 837 838(* ::Subsubsection:: *) 839(*Program code:*) 840 841 842(* ::Code:: *) 843Int[(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 844 (-c*Cos[d+e*x]+b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1)/(e*n) + 845 Dist[a*(2*n-1)/n,Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1),x]] /; 846FreeQ[{a,b,c,d,e},x] && ZeroQ[a^2-b^2-c^2] && RationalQ[n] && n>1 847 848 849(* ::Subsubsection:: *) 850(**) 851 852 853(* ::Subsubsection:: *) 854(*Reference: G&R 2.558.1 inverted*) 855 856 857(* ::Subsubsection:: *) 858(*Rule: If a^2-b^2-c^2!=0 \[And] n\[Element]\[DoubleStruckCapitalF] \[And] n>1, then*) 859 860 861(* ::Subsubtitle::Bold:: *) 862(*\[Integral](a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x \[LongRightArrow] *) 863(* (((-c Cos[d+e x]+b Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^(n-1))/(e n))+*) 864(*1/n \[Integral](n a^2+(n-1)(b^2+c^2)+a b (2 n-1) Cos[d+e x]+a c (2 n-1)Sin[d+e x])(a+b Cos[d+e x]+c Sin[d+e x])^(n-2) \[DifferentialD]x*) 865 866 867(* ::Subsubsection:: *) 868(*Program code:*) 869 870 871(* ::Code:: *) 872Int[(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 873 (-c*Cos[d+e*x]+b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1)/(e*n) + 874 Dist[1/n,Int[(n*a^2+(n-1)*(b^2+c^2)+a*b*(2*n-1)*Cos[d+e*x]+a*c*(2*n-1)*Sin[d+e*x])* 875 (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-2),x]] /; 876FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2-c^2] && FractionQ[n] && n>1 877 878 879(* ::PageBreak:: *) 880(**) 881 882 883(* ::Title::Bold::Closed:: *) 884(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x*) 885 886 887(* ::Subsubsection:: *) 888(*Reference: G&R 2.558.1 inverted with a^2-b^2-c^2=0 inverted*) 889 890 891(* ::Subsubsection:: *) 892(*Rule: If a^2-b^2-c^2=0 \[And] n<-1, then*) 893 894 895(* ::Subsubtitle::Bold:: *) 896(*\[Integral](a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x \[LongRightArrow] *) 897(* (((c Cos[d+e x]-b Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^n)/(a e (2 n+1)))+*) 898(*(n+1)/(a (2 n+1)) \[Integral](a+b Cos[d+e x]+c Sin[d+e x])^(n+1) \[DifferentialD]x*) 899 900 901(* ::Subsubsection:: *) 902(*Program code:*) 903 904 905(* ::Code:: *) 906Int[(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 907 (c*Cos[d+e*x]-b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(2*n+1)) + 908 Dist[(n+1)/(a*(2*n+1)),Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x]] /; 909FreeQ[{a,b,c,d,e},x] && ZeroQ[a^2-b^2-c^2] && RationalQ[n] && n<-1 910 911 912(* ::Subsubsection:: *) 913(**) 914 915 916(* ::Subsubsection:: *) 917(*Reference: G&R 2.558.1 with n=-2*) 918 919 920(* ::Subsubsection:: *) 921(*Rule: If a^2-b^2-c^2!=0, then*) 922 923 924(* ::Subsubtitle::Bold:: *) 925(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x])^2 \[DifferentialD]x \[LongRightArrow] *) 926(* ((c Cos[d+e x]-b Sin[d+e x])/(e (a^2-b^2-c^2) (a+b Cos[d+e x]+c Sin[d+e x])))+a/(a^2-b^2-c^2) \[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x*) 927 928 929(* ::Subsubsection:: *) 930(*Program code:*) 931 932 933(* ::Code:: *) 934Int[1/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^2,x_Symbol] := 935 (c*Cos[d+e*x]-b*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) + 936 Dist[a/(a^2-b^2-c^2),Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x]] /; 937FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2-c^2] 938 939 940(* ::Subsubsection:: *) 941(**) 942 943 944(* ::Subsubsection:: *) 945(*Reference: G&R 2.558.1 with n=-(3/2)*) 946 947 948(* ::Subsubsection:: *) 949(*Rule: If a^2-b^2-c^2!=0, then*) 950 951 952(* ::Subsubtitle::Bold:: *) 953(*\[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x])^(3/2) \[DifferentialD]x \[LongRightArrow] *) 954(* ((2 (c Cos[d+e x]-b Sin[d+e x]))/(e (a^2-b^2-c^2) Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]))+1/(a^2-b^2-c^2) \[Integral]Sqrt[a+b Cos[d+e x]+c Sin[d+e x]]\[DifferentialD]x*) 955 956 957(* ::Subsubsection:: *) 958(*Program code:*) 959 960 961(* ::Code:: *) 962Int[1/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^(3/2),x_Symbol] := 963 2*(c*Cos[d+e*x]-b*Sin[d+e*x])/(e*(a^2-b^2-c^2)*Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]]) + 964 Dist[1/(a^2-b^2-c^2),Int[Sqrt[a+b*Cos[d+e*x]+c*Sin[d+e*x]],x]] /; 965FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2-c^2] 966 967 968(* ::Subsubsection:: *) 969(**) 970 971 972(* ::Subsubsection:: *) 973(*Reference: G&R 2.558.1*) 974 975 976(* ::Subsubsection:: *) 977(*Rule: If a^2-b^2-c^2!=0 \[And] n<-1 \[And] n!=-2 \[And] n!=-(3/2), then*) 978 979 980(* ::Subsubtitle::Bold:: *) 981(*\[Integral](a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x \[LongRightArrow] (((-c Cos[d+e x]+b Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^(n+1))/(e (n+1) (a^2-b^2-c^2)))+*) 982(*1/((n+1) (a^2-b^2-c^2)) \[Integral]((n+1) a-(n+2) b Cos[d+e x]-(n+2) c Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^(n+1) \[DifferentialD]x*) 983 984 985(* ::Subsubsection:: *) 986(*Program code:*) 987 988 989(* ::Code:: *) 990Int[(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 991 (-c*Cos[d+e*x]+b*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/(e*(n+1)*(a^2-b^2-c^2)) + 992 Dist[1/((n+1)*(a^2-b^2-c^2)), 993 Int[((n+1)*a-(n+2)*b*Cos[d+e*x]-(n+2)*c*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x]] /; 994FreeQ[{a,b,c,d,e},x] && NonzeroQ[a^2-b^2-c^2] && RationalQ[n] && n<-1 && n!=-2 && n!=-3/2 995 996 997(* ::PageBreak:: *) 998(**) 999 1000 1001(* ::Title::Bold::Closed:: *) 1002(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])(a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x*) 1003 1004 1005(* ::Subsubsection:: *) 1006(*Note: Although exactly analogous to G&R 2.451.3 for hyperbolic functions, there is no corresponding G&R 2.558.n formula for trig functions. Apparently the authors did not anticipate b^2+c^2 could be 0 in the complex plane.*) 1007 1008 1009(* ::Subsubsection:: *) 1010(*Rule: If b^2+c^2=0, then*) 1011 1012 1013(* ::Subsubtitle::Bold:: *) 1014(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] (((2 a A-b B-c C) x)/(2 a^2))-((b B+c C) (b Cos[d+e x]-c Sin[d+e x]))/(2 a b c e)+*) 1015(*((a^2 (b B-c C)-2 a A b^2+b^2 (b B+c C)) Log[a+b Cos[d+e x]+c Sin[d+e x]])/(2 a^2 b c e)*) 1016 1017 1018(* ::Subsubsection:: *) 1019(*Program code:*) 1020 1021 1022(* ::Code:: *) 1023Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1024 (2*a*A-b*B-c*C)*x/(2*a^2) - (b*B+c*C)*(b*Cos[d+e*x]-c*Sin[d+e*x])/(2*a*b*c*e) + 1025 (a^2*(b*B-c*C)-2*a*A*b^2+b^2*(b*B+c*C))*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(2*a^2*b*c*e) /; 1026FreeQ[{a,b,c,d,e,A,B,C},x] && ZeroQ[b^2+c^2] 1027 1028 1029(* ::Code:: *) 1030Int[(A_.+C_.*Sin[d_.+e_.*x_])/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1031 (2*a*A-c*C)*x/(2*a^2) - C*Cos[d+e*x]/(2*a*e) + c*C*Sin[d+e*x]/(2*a*b*e) + 1032 (-a^2*C+2*a*c*A+b^2*C)*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(2*a^2*b*e) /; 1033FreeQ[{a,b,c,d,e,A,C},x] && ZeroQ[b^2+c^2] 1034 1035 1036(* ::Code:: *) 1037Int[(A_.+B_.*Cos[d_.+e_.*x_])/(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1038 (2*a*A-b*B)*x/(2*a^2) - b*B*Cos[d+e*x]/(2*a*c*e) + B*Sin[d+e*x]/(2*a*e) + 1039 (a^2*B-2*a*b*A+b^2*B)*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(2*a^2*c*e) /; 1040FreeQ[{a,b,c,d,e,A,B},x] && ZeroQ[b^2+c^2] 1041 1042 1043(* ::Subsubsection:: *) 1044(**) 1045 1046 1047(* ::Subsubsection:: *) 1048(*Reference: G&R 2.558.2 with A(b^2+c^2)-a(b B+c C)=0*) 1049 1050 1051(* ::Subsubsection:: *) 1052(*Rule: If b^2+c^2!=0 \[And] A(b^2+c^2)-a(b B+c C)=0, then*) 1053 1054 1055(* ::Subsubtitle::Bold:: *) 1056(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] (((b B+c C) x)/(b^2+c^2))+((c B-b C) Log[a+b Cos[d+e x]+c Sin[d+e x]])/(e (b^2+c^2))*) 1057 1058 1059(* ::Subsubsection:: *) 1060(*Program code:*) 1061 1062 1063(* ::Code:: *) 1064Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1065 (b*B+c*C)*x/(b^2+c^2) + (c*B-b*C)*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) /; 1066FreeQ[{a,b,c,d,e,A,B,C},x] && NonzeroQ[b^2+c^2] && ZeroQ[A*(b^2+c^2)-a*(b*B+c*C)] 1067 1068 1069(* ::Subsubsection:: *) 1070(*Reference: G&R 2.558.2 with B=0 and A (b^2+c^2)-a c C=0*) 1071 1072 1073(* ::Code:: *) 1074Int[(A_.+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1075 c*C*x/(b^2+c^2) - b*C*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) /; 1076FreeQ[{a,b,c,d,e,A,C},x] && NonzeroQ[b^2+c^2] && ZeroQ[A*(b^2+c^2)-a*c*C] 1077 1078 1079(* ::Subsubsection:: *) 1080(*Reference: G&R 2.558.2 with C=0 and A (b^2+c^2)-a b B=0*) 1081 1082 1083(* ::Code:: *) 1084Int[(A_.+B_.*Cos[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1085 b*B*x/(b^2+c^2) + c*B*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) /; 1086FreeQ[{a,b,c,d,e,A,B},x] && NonzeroQ[b^2+c^2] && ZeroQ[A*(b^2+c^2)-a*b*B] 1087 1088 1089(* ::Subsubsection:: *) 1090(**) 1091 1092 1093(* ::Subsubsection:: *) 1094(*Reference: G&R 2.558.2*) 1095 1096 1097(* ::Subsubsection:: *) 1098(*Rule: If b^2+c^2!=0 \[And] A (b^2+c^2)-a (b B+c C)!=0, then*) 1099 1100 1101(* ::Subsubtitle::Bold:: *) 1102(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x \[LongRightArrow] (((b B+c C) x)/(b^2+c^2))+*) 1103(*((c B-b C) Log[a+b Cos[d+e x]+c Sin[d+e x]])/(e (b^2+c^2))+(A (b^2+c^2)-a (b B+c C))/(b^2+c^2) \[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x*) 1104 1105 1106(* ::Subsubsection:: *) 1107(*Program code:*) 1108 1109 1110(* ::Code:: *) 1111Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1112 (b*B+c*C)*x/(b^2+c^2) + (c*B-b*C)*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) + 1113 Dist[(A*(b^2+c^2)-a*(b*B+c*C))/(b^2+c^2),Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x]] /; 1114FreeQ[{a,b,c,d,e,A,B,C},x] && NonzeroQ[b^2+c^2] && NonzeroQ[A*(b^2+c^2)-a*(b*B+c*C)] 1115 1116 1117(* ::Subsubsection:: *) 1118(*Reference: G&R 2.558.2 with B=0*) 1119 1120 1121(* ::Code:: *) 1122Int[(A_.+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1123 c*C*(d+e*x)/(e*(b^2+c^2)) - b*C*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) + 1124 Dist[(A*(b^2+c^2)-a*c*C)/(b^2+c^2),Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x]] /; 1125FreeQ[{a,b,c,d,e,A,C},x] && NonzeroQ[b^2+c^2] && NonzeroQ[A*(b^2+c^2)-a*c*C] 1126 1127 1128(* ::Subsubsection:: *) 1129(*Reference: G&R 2.558.2 with C=0*) 1130 1131 1132(* ::Code:: *) 1133Int[(A_.+B_.*Cos[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_]),x_Symbol] := 1134 b*B*(d+e*x)/(e*(b^2+c^2)) + 1135 c*B*Log[a+b*Cos[d+e*x]+c*Sin[d+e*x]]/(e*(b^2+c^2)) + 1136 Dist[(A*(b^2+c^2)-a*b*B)/(b^2+c^2),Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x]] /; 1137FreeQ[{a,b,c,d,e,A,B},x] && NonzeroQ[b^2+c^2] && NonzeroQ[A*(b^2+c^2)-a*b*B] 1138 1139 1140(* ::Subsubsection:: *) 1141(**) 1142 1143 1144(* ::Subsubsection:: *) 1145(*Reference: G&R 2.558.1 with n=-2 and a A-b B-c C=0*) 1146 1147 1148(* ::Subsubsection:: *) 1149(*Rule: If a^2-b^2-c^2!=0 \[And] a A-b B-c C=0, then*) 1150 1151 1152(* ::Subsubtitle::Bold:: *) 1153(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])/(a+b Cos[d+e x]+c Sin[d+e x])^2 \[DifferentialD]x \[LongRightArrow] ((c B-b C-(a C-c A) Cos[d+e x]+(a B-b A) Sin[d+e x])/(e (a^2-b^2-c^2) (a+b Cos[d+e x]+c Sin[d+e x])))*) 1154 1155 1156(* ::Subsubsection:: *) 1157(*Program code:*) 1158 1159 1160(* ::Code:: *) 1161Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^2,x_Symbol] := 1162 (c*B-b*C-(a*C-c*A)*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/ 1163 (e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) /; 1164FreeQ[{a,b,c,d,e,A,B,C},x] && NonzeroQ[a^2-b^2-c^2] && ZeroQ[a*A-b*B-c*C] 1165 1166 1167(* ::Subsubsection:: *) 1168(*Reference: G&R 2.558.1 with B=0, n=-2 and a A-c C=0*) 1169 1170 1171(* ::Code:: *) 1172Int[(A_.+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^2,x_Symbol] := 1173 -(b*C+(a*C-c*A)*Cos[d+e*x]+b*A*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) /; 1174FreeQ[{a,b,c,d,e,A,C},x] && NonzeroQ[a^2-b^2-c^2] && ZeroQ[a*A-c*C] 1175 1176 1177(* ::Subsubsection:: *) 1178(*Reference: G&R 2.558.1 with C=0, n=-2 and a A-b B=0*) 1179 1180 1181(* ::Code:: *) 1182Int[(A_.+B_.*Cos[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^2,x_Symbol] := 1183 (c*B+c*A*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) /; 1184FreeQ[{a,b,c,d,e,A,B},x] && NonzeroQ[a^2-b^2-c^2] && ZeroQ[a*A-b*B] 1185 1186 1187(* ::Subsubsection:: *) 1188(**) 1189 1190 1191(* ::Subsubsection:: *) 1192(*Reference: G&R 2.558.1 with n=-2*) 1193 1194 1195(* ::Subsubsection:: *) 1196(*Rule: If a^2-b^2-c^2!=0 \[And] a A-b B-c C!=0, then*) 1197 1198 1199(* ::Subsubtitle::Bold:: *) 1200(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])/(a+b Cos[d+e x]+c Sin[d+e x])^2 \[DifferentialD]x \[LongRightArrow] *) 1201(*((c B-b C-(a C-c A) Cos[d+e x]+(a B-b A) Sin[d+e x])/(e (a^2-b^2-c^2) (a+b Cos[d+e x]+c Sin[d+e x])))+(a A-b B-c C)/(a^2-b^2-c^2) \[Integral]1/(a+b Cos[d+e x]+c Sin[d+e x]) \[DifferentialD]x*) 1202 1203 1204(* ::Subsubsection:: *) 1205(*Program code:*) 1206 1207 1208(* ::Code:: *) 1209Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^2,x_Symbol] := 1210 (c*B-b*C-(a*C-c*A)*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/ 1211 (e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) + 1212 Dist[(a*A-b*B-c*C)/(a^2-b^2-c^2),Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x]] /; 1213FreeQ[{a,b,c,d,e,A,B,C},x] && NonzeroQ[a^2-b^2-c^2] && NonzeroQ[a*A-b*B-c*C] 1214 1215 1216(* ::Subsubsection:: *) 1217(*Reference: G&R 2.558.1 with B=0 and n=-2*) 1218 1219 1220(* ::Code:: *) 1221Int[(A_.+C_.*Sin[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^2,x_Symbol] := 1222 -(b*C+(a*C-c*A)*Cos[d+e*x]+b*A*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) + 1223 Dist[(a*A-c*C)/(a^2-b^2-c^2),Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x]] /; 1224FreeQ[{a,b,c,d,e,A,C},x] && NonzeroQ[a^2-b^2-c^2] && NonzeroQ[a*A-c*C] 1225 1226 1227(* ::Subsubsection:: *) 1228(*Reference: G&R 2.558.1 with C=0 and n=-2*) 1229 1230 1231(* ::Code:: *) 1232Int[(A_.+B_.*Cos[d_.+e_.*x_])/(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^2,x_Symbol] := 1233 (c*B+c*A*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])/(e*(a^2-b^2-c^2)*(a+b*Cos[d+e*x]+c*Sin[d+e*x])) + 1234 Dist[(a*A-b*B)/(a^2-b^2-c^2),Int[1/(a+b*Cos[d+e*x]+c*Sin[d+e*x]),x]] /; 1235FreeQ[{a,b,c,d,e,A,B},x] && NonzeroQ[a^2-b^2-c^2] && NonzeroQ[a*A-b*B] 1236 1237 1238(* ::Subsubsection:: *) 1239(**) 1240 1241 1242(* ::Subsubsection:: *) 1243(*Reference: G&R 2.558.1*) 1244 1245 1246(* ::Subsubsection:: *) 1247(*Rule: If a^2-b^2-c^2!=0 \[And] n<-1 \[And] n!=-2, then*) 1248 1249 1250(* ::Subsubtitle::Bold:: *) 1251(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])(a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x \[LongRightArrow] *) 1252(*-(((c B-b C-(a C-c A) Cos[d+e x]+(a B-b A) Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^(n+1))/(e (n+1) (a^2-b^2-c^2)))+*) 1253(*1/((n+1) (a^2-b^2-c^2)) \[Integral]((n+1) (a A-b B-c C)+(n+2) (a B-b A) Cos[d+e x]+(n+2) (a C-c A) Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^(n+1) \[DifferentialD]x*) 1254 1255 1256(* ::Subsubsection:: *) 1257(*Program code:*) 1258 1259 1260(* ::Code:: *) 1261Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])*(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 1262 -(c*B-b*C-(a*C-c*A)*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/ 1263 (e*(n+1)*(a^2-b^2-c^2)) + 1264 Dist[1/((n+1)*(a^2-b^2-c^2)), 1265 Int[((n+1)*(a*A-b*B-c*C)+(n+2)*(a*B-b*A)*Cos[d+e*x]+(n+2)*(a*C-c*A)*Sin[d+e*x])* 1266 (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x]] /; 1267FreeQ[{a,b,c,d,e,A,B,C},x] && NonzeroQ[a^2-b^2-c^2] && RationalQ[n] && n<-1 && n!=-2 1268 1269 1270(* ::Subsubsection:: *) 1271(*Reference: G&R 2.558.1 with B=0*) 1272 1273 1274(* ::Code:: *) 1275Int[(A_.+C_.*Sin[d_.+e_.*x_])*(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 1276 (b*C+(a*C-c*A)*Cos[d+e*x]+b*A*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/ 1277 (e*(n+1)*(a^2-b^2-c^2)) + 1278 Dist[1/((n+1)*(a^2-b^2-c^2)), 1279 Int[((n+1)*(a*A-c*C)-(n+2)*b*A*Cos[d+e*x]+(n+2)*(a*C-c*A)*Sin[d+e*x])* 1280 (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x]] /; 1281FreeQ[{a,b,c,d,e,A,C},x] && NonzeroQ[a^2-b^2-c^2] && RationalQ[n] && n<-1 && n!=-2 1282 1283 1284(* ::Subsubsection:: *) 1285(*Reference: G&R 2.558.1 with C=0*) 1286 1287 1288(* ::Code:: *) 1289Int[(A_.+B_.*Cos[d_.+e_.*x_])*(a_.+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 1290 -(c*B+c*A*Cos[d+e*x]+(a*B-b*A)*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1)/ 1291 (e*(n+1)*(a^2-b^2-c^2)) + 1292 Dist[1/((n+1)*(a^2-b^2-c^2)), 1293 Int[((n+1)*(a*A-b*B)+(n+2)*(a*B-b*A)*Cos[d+e*x]-(n+2)*c*A*Sin[d+e*x])* 1294 (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x]] /; 1295FreeQ[{a,b,c,d,e,A,B},x] && NonzeroQ[a^2-b^2-c^2] && RationalQ[n] && n<-1 && n!=-2 1296 1297 1298(* ::Subsubsection:: *) 1299(**) 1300 1301 1302(* ::Subsubsection:: *) 1303(*Derivation: Algebraic simplification*) 1304 1305 1306(* ::Subsubsection:: *) 1307(*Basis: (A+B z) (a+b z)^n=B/b (a+b z)^(n+1)+(A b-a B)/b (a+b z)^n*) 1308 1309 1310(* ::Subsubsection:: *) 1311(*Rule: If b C-c B=0 \[And] b A-a B!=0 \[And] (n=-(1/2) \[Or] a^2-b^2-c^2=0), then*) 1312 1313 1314(* ::Subsubtitle::Bold:: *) 1315(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])(a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x \[LongRightArrow] *) 1316(* (B/b)\[Integral](a+b Cos[d+e x]+c Sin[d+e x])^(n+1) \[DifferentialD]x+(b A-a B)/b \[Integral](a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x*) 1317 1318 1319(* ::Subsubsection:: *) 1320(*Program code:*) 1321 1322 1323(* ::Code:: *) 1324Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])*(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 1325 Dist[B/b,Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n+1),x]] + 1326 Dist[(b*A-a*B)/b,Int[(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n,x]] /; 1327FreeQ[{a,b,c,d,e,A,B,C},x] && ZeroQ[b*C-c*B] && NonzeroQ[b*A-a*B] && RationalQ[n] && (n==-1/2 || ZeroQ[a^2-b^2-c^2]) 1328 1329 1330(* ::Subsubsection:: *) 1331(**) 1332 1333 1334(* ::Subsubsection:: *) 1335(*Reference: G&R 2.558.1 inverted*) 1336 1337 1338(* ::Subsubsection:: *) 1339(*Rule: If a^2-b^2-c^2!=0 \[And] n\[Element]\[DoubleStruckCapitalF] \[And] n>0, then*) 1340 1341 1342(* ::Subsubtitle::Bold:: *) 1343(*\[Integral](A+B Cos[d+e x]+C Sin[d+e x])(a+b Cos[d+e x]+c Sin[d+e x])^n \[DifferentialD]x \[LongRightArrow] *) 1344(*(((B c-b C-a C Cos[d+e x]+a B Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^n)/(a e (n+1)))+*) 1345(*1/(a (n+1)) \[Integral](a (b B+c C) n+a^2 A (n+1)+(a^2 B n+c (b C-c B) n+a b A (n+1)) Cos[d+e x]+(a^2 C n-b (b C-c B) n+a c A (n+1)) Sin[d+e x]) (a+b Cos[d+e x]+c Sin[d+e x])^(n-1) \[DifferentialD]x*) 1346 1347 1348(* ::Subsubsection:: *) 1349(*Program code:*) 1350 1351 1352(* ::Code:: *) 1353Int[(A_.+B_.*Cos[d_.+e_.*x_]+C_.*Sin[d_.+e_.*x_])*(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 1354 (B*c-b*C-a*C*Cos[d+e*x]+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + 1355 Dist[1/(a*(n+1)), 1356 Int[(a*(b*B+c*C)*n + a^2*A*(n+1) + 1357 (a^2*B*n + c*(b*C-c*B)*n + a*b*A*(n+1))*Cos[d+e*x] + 1358 (a^2*C*n - b*(b*C-c*B)*n + a*c*A*(n+1))*Sin[d+e*x])* 1359 (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1),x]] /; 1360FreeQ[{a,b,c,d,e,A,B,C},x] && NonzeroQ[a^2-b^2-c^2] && RationalQ[n] && n>0 1361 1362 1363(* ::Subsubsection:: *) 1364(*Reference: G&R 2.558.1 inverted with B=0*) 1365 1366 1367(* ::Code:: *) 1368Int[(A_.+C_.*Sin[d_.+e_.*x_])*(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 1369 -(b*C+a*C*Cos[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + 1370 Dist[1/(a*(n+1)), 1371 Int[(a*c*C*n+a^2*A*(n+1)+(c*b*C*n+a*b*A*(n+1))*Cos[d+e*x]+(a^2*C*n-b^2*C*n+a*c*A*(n+1))*Sin[d+e*x])* 1372 (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1),x]] /; 1373FreeQ[{a,b,c,d,e,A,C},x] && NonzeroQ[a^2-b^2-c^2] && RationalQ[n] && n>0 1374 1375 1376(* ::Subsubsection:: *) 1377(*Reference: G&R 2.558.1 inverted with C=0*) 1378 1379 1380(* ::Code:: *) 1381Int[(A_.+B_.*Cos[d_.+e_.*x_])*(a_+b_.*Cos[d_.+e_.*x_]+c_.*Sin[d_.+e_.*x_])^n_,x_Symbol] := 1382 (B*c+a*B*Sin[d+e*x])*(a+b*Cos[d+e*x]+c*Sin[d+e*x])^n/(a*e*(n+1)) + 1383 Dist[1/(a*(n+1)), 1384 Int[(a*b*B*n+a^2*A*(n+1)+(a^2*B*n-c^2*B*n+a*b*A*(n+1))*Cos[d+e*x]+(b*c*B*n+a*c*A*(n+1))*Sin[d+e*x])* 1385 (a+b*Cos[d+e*x]+c*Sin[d+e*x])^(n-1),x]] /; 1386FreeQ[{a,b,c,d,e,A,B},x] && NonzeroQ[a^2-b^2-c^2] && RationalQ[n] && n>0 1387 1388 1389(* ::PageBreak:: *) 1390(**) 1391