1(* ::Package:: *) 2 3(* ::Title::Bold::Closed:: *) 4(*\[Integral]Tanh[a+b x]^n \[DifferentialD]x*) 5 6 7(* ::Subsubsection:: *) 8(*Reference: G&R 2.243.17, CRC 556, A&S 4.5.79*) 9 10 11(* ::Subsubsection:: *) 12(*Derivation: Reciprocal rule*) 13 14 15(* ::Subsubsection:: *) 16(*Basis: Tanh[z]=Sinh[z]/Cosh[z]*) 17 18 19(* ::Subsubsection:: *) 20(*Rule:*) 21 22 23(* ::Subsubtitle::Bold:: *) 24(*\[Integral]Tanh[a+b x]\[DifferentialD]x \[LongRightArrow] (Log[Cosh[a+b x]]/b)*) 25 26 27(* ::Subsubsection:: *) 28(*Program code:*) 29 30 31(* ::Code:: *) 32Int[Tanh[a_.+b_.*x_],x_Symbol] := 33 Log[Cosh[a+b*x]]/b /; 34FreeQ[{a,b},x] 35 36 37(* ::Subsubsection:: *) 38(*Reference: G&R 2.423.33, CRC 557, A&S 4.5.82*) 39 40 41(* ::Code:: *) 42Int[Coth[a_.+b_.*x_],x_Symbol] := 43 Log[Sinh[a+b*x]]/b /; 44FreeQ[{a,b},x] 45 46 47(* ::Subsubsection:: *) 48(**) 49 50 51(* ::Subsubsection:: *) 52(*Reference: G&R 2.423.22, CRC 569*) 53 54 55(* ::Subsubsection:: *) 56(*Derivation: Algebraic expansion*) 57 58 59(* ::Subsubsection:: *) 60(*Basis: Tanh[z]^2=1-Sech[z]^2*) 61 62 63(* ::Subsubsection:: *) 64(*Rule:*) 65 66 67(* ::Subsubtitle::Bold:: *) 68(*\[Integral]Tanh[a+b x]^2 \[DifferentialD]x \[LongRightArrow] x-Tanh[a+b x]/b*) 69 70 71(* ::Subsubsection:: *) 72(*Program code:*) 73 74 75(* ::Code:: *) 76Int[Tanh[a_.+b_.*x_]^2,x_Symbol] := 77 x - Tanh[a+b*x]/b /; 78FreeQ[{a,b},x] 79 80 81(* ::Subsubsection:: *) 82(*Reference: G&R 2.423.38, CRC 573*) 83 84 85(* ::Code:: *) 86Int[Coth[a_.+b_.*x_]^2,x_Symbol] := 87 x - Coth[a+b*x]/b /; 88FreeQ[{a,b},x] 89 90 91(* ::Subsubsection:: *) 92(**) 93 94 95(* ::Subsubsection:: *) 96(*Reference: G&R 2.411.3, CRC 570, A&S 4.5.87*) 97 98 99(* ::Subsubsection:: *) 100(*Derivation: Integration by parts with a double-back flip*) 101 102 103(* ::Subsubsection:: *) 104(*Basis: Tanh[z]^n=(Tanh[z]^(n-1) Sinh[z])/Cosh[z]*) 105 106 107(* ::Subsubsection:: *) 108(*Rule: If n>1, then*) 109 110 111(* ::Subsubtitle::Bold:: *) 112(*\[Integral](c Tanh[a+b x])^n \[DifferentialD]x \[LongRightArrow] -((c (c Tanh[a+b x])^(n-1))/(b (n-1)))+c^2 \[Integral](c Tanh[a+b x])^(n-2) \[DifferentialD]x*) 113 114 115(* ::Subsubsection:: *) 116(*Program code:*) 117 118 119(* ::Code:: *) 120Int[(c_.*Tanh[a_.+b_.*x_])^n_,x_Symbol] := 121 -c*(c*Tanh[a+b*x])^(n-1)/(b*(n-1)) + 122 Dist[c^2,Int[(c*Tanh[a+b*x])^(n-2),x]] /; 123FreeQ[{a,b,c},x] && RationalQ[n] && n>1 124 125 126(* ::Subsubsection:: *) 127(*Reference: G&R 2.411.4, CRC 574, A&S 4.5.88*) 128 129 130(* ::Code:: *) 131Int[(c_.*Coth[a_.+b_.*x_])^n_,x_Symbol] := 132 -c*(c*Coth[a+b*x])^(n-1)/(b*(n-1)) + 133 Dist[c^2,Int[(c*Coth[a+b*x])^(n-2),x]] /; 134FreeQ[{a,b,c},x] && RationalQ[n] && n>1 135 136 137(* ::Subsubsection:: *) 138(**) 139 140 141(* ::Subsubsection:: *) 142(*Reference: G&R 2.411.4, CRC 574'*) 143 144 145(* ::Subsubsection:: *) 146(*Derivation: Inverted integration by parts with a double-back flip*) 147 148 149(* ::Subsubsection:: *) 150(*Rule: If n<-1, then*) 151 152 153(* ::Subsubtitle::Bold:: *) 154(*\[Integral](c Tanh[a+b x])^n \[DifferentialD]x \[LongRightArrow] ((c Tanh[a+b x])^(n+1)/(b c (n+1)))+1/c^2 \[Integral](c Tanh[a+b x])^(n+2) \[DifferentialD]x*) 155 156 157(* ::Subsubsection:: *) 158(*Program code:*) 159 160 161(* ::Code:: *) 162Int[(c_.*Tanh[a_.+b_.*x_])^n_,x_Symbol] := 163 (c*Tanh[a+b*x])^(n+1)/(b*c*(n+1)) + 164 Dist[1/c^2,Int[(c*Tanh[a+b*x])^(n+2),x]] /; 165FreeQ[{a,b,c},x] && RationalQ[n] && n<-1 166 167 168(* ::Subsubsection:: *) 169(*Reference: G&R 2.411.3, CRC 570'*) 170 171 172(* ::Code:: *) 173Int[(c_.*Coth[a_.+b_.*x_])^n_,x_Symbol] := 174 (c*Coth[a+b*x])^(n+1)/(b*c*(n+1)) + 175 Dist[1/c^2,Int[(c*Coth[a+b*x])^(n+2),x]] /; 176FreeQ[{a,b,c},x] && RationalQ[n] && n<-1 177 178 179(* ::PageBreak:: *) 180(**) 181 182 183(* ::Title::Bold::Closed:: *) 184(*\[Integral](a+b Tanh[c+d x])^n \[DifferentialD]x when a^2-b^2=0*) 185 186 187(* ::Subsubsection:: *) 188(*Rule: If a^2-b^2=0, then*) 189 190 191(* ::Subsubtitle::Bold:: *) 192(*\[Integral]1/(a+b Tanh[c+d x]) \[DifferentialD]x \[LongRightArrow] (x/(2 a))-a/(2 b d (a+b Tanh[c+d x]))*) 193 194 195(* ::Subsubsection:: *) 196(*Program code:*) 197 198 199(* ::Code:: *) 200Int[1/(a_+b_.*Tanh[c_.+d_.*x_]),x_Symbol] := 201 x/(2*a) - a/(2*b*d*(a+b*Tanh[c+d*x])) /; 202FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] 203 204 205(* ::Code:: *) 206Int[1/(a_+b_.*Coth[c_.+d_.*x_]),x_Symbol] := 207 x/(2*a) - a/(2*b*d*(a+b*Coth[c+d*x])) /; 208FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] 209 210 211(* ::Subsubsection:: *) 212(**) 213 214 215(* ::Subsubsection:: *) 216(*Rule: If a^2-b^2=0 \[And] a>0, then*) 217 218 219(* ::Subsubtitle::Bold:: *) 220(*\[Integral]Sqrt[a+b Tanh[c+d x]]\[DifferentialD]x \[LongRightArrow] ((Sqrt[2] b)/(d Sqrt[a]))ArcTanh[Sqrt[a+b Tanh[c+d x]]/(Sqrt[2] Sqrt[a])]*) 221 222 223(* ::Subsubsection:: *) 224(*Program code:*) 225 226 227(* ::Code:: *) 228Int[Sqrt[a_+b_.*Tanh[c_.+d_.*x_]],x_Symbol] := 229 Sqrt[2]*b/(d*Rt[a,2])*ArcTanh[Sqrt[a+b*Tanh[c+d*x]]/(Sqrt[2]*Rt[a,2])] /; 230FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && PosQ[a] 231 232 233(* ::Code:: *) 234Int[Sqrt[a_+b_.*Coth[c_.+d_.*x_]],x_Symbol] := 235 (Sqrt[2]*b/(d*Rt[a,2])*ArcCoth[Sqrt[a+b*Coth[c+d*x]]/(Sqrt[2]*Rt[a,2])]) /; 236FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && PosQ[a] 237 238 239(* ::Subsubsection:: *) 240(**) 241 242 243(* ::Subsubsection:: *) 244(*Rule: If a^2-b^2=0 \[And] \[Not](a>0), then*) 245 246 247(* ::Subsubtitle::Bold:: *) 248(*\[Integral]Sqrt[a+b Tanh[c+d x]]\[DifferentialD]x \[LongRightArrow] -((Sqrt[2] b)/(d Sqrt[-a]))ArcTan[Sqrt[a+b Tanh[c+d x]]/(Sqrt[2] Sqrt[-a])]*) 249 250 251(* ::Subsubsection:: *) 252(*Program code:*) 253 254 255(* ::Code:: *) 256Int[Sqrt[a_+b_.*Tanh[c_.+d_.*x_]],x_Symbol] := 257 -Sqrt[2]*b/(d*Rt[-a,2])*ArcTan[Sqrt[a+b*Tanh[c+d*x]]/(Sqrt[2]*Rt[-a,2])] /; 258FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && NegQ[a] 259 260 261(* ::Code:: *) 262Int[Sqrt[a_+b_.*Coth[c_.+d_.*x_]],x_Symbol] := 263 Sqrt[2]*b/(d*Rt[-a,2])*ArcCot[Sqrt[a+b*Coth[c+d*x]]/(Sqrt[2]*Rt[-a,2])] /; 264FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && NegQ[a] 265 266 267(* ::Subsubsection:: *) 268(**) 269 270 271(* ::Subsubsection:: *) 272(*Rule: If a^2-b^2=0 \[And] n\[Element]\[DoubleStruckCapitalF] \[And] n>1, then*) 273 274 275(* ::Subsubtitle::Bold:: *) 276(*\[Integral](a+b Tanh[c+d x])^n \[DifferentialD]x \[LongRightArrow] -((a^2 (a+b Tanh[c+d x])^(n-1))/(b d (n-1)))+2 a \[Integral](a+b Tanh[c+d x])^(n-1) \[DifferentialD]x*) 277 278 279(* ::Subsubsection:: *) 280(*Program code:*) 281 282 283(* ::Code:: *) 284Int[(a_+b_.*Tanh[c_.+d_.*x_])^n_,x_Symbol] := 285 -a^2*(a+b*Tanh[c+d*x])^(n-1)/(b*d*(n-1)) + 286 Dist[2*a,Int[(a+b*Tanh[c+d*x])^(n-1),x]] /; 287FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && FractionQ[n] && n>1 288 289 290(* ::Code:: *) 291Int[(a_+b_.*Coth[c_.+d_.*x_])^n_,x_Symbol] := 292 -a^2*(a+b*Coth[c+d*x])^(n-1)/(b*d*(n-1)) + 293 Dist[2*a,Int[(a+b*Coth[c+d*x])^(n-1),x]] /; 294FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && FractionQ[n] && n>1 295 296 297(* ::Subsubsection:: *) 298(**) 299 300 301(* ::Subsubsection:: *) 302(*Rule: If a^2-b^2=0 \[And] n<0, then*) 303 304 305(* ::Subsubtitle::Bold:: *) 306(*\[Integral](a+b Tanh[c+d x])^n \[DifferentialD]x \[LongRightArrow] ((a (a+b Tanh[c+d x])^n)/(2 b d n))+1/(2 a) \[Integral](a+b Tanh[c+d x])^(n+1) \[DifferentialD]x*) 307 308 309(* ::Subsubsection:: *) 310(*Program code:*) 311 312 313(* ::Code:: *) 314Int[(a_+b_.*Tanh[c_.+d_.*x_])^n_,x_Symbol] := 315 a*(a+b*Tanh[c+d*x])^n/(2*b*d*n) + 316 Dist[1/(2*a),Int[(a+b*Tanh[c+d*x])^(n+1),x]] /; 317FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && RationalQ[n] && n<0 318 319 320(* ::Code:: *) 321Int[(a_+b_.*Coth[c_.+d_.*x_])^n_,x_Symbol] := 322 a*(a+b*Coth[c+d*x])^n/(2*b*d*n) + 323 Dist[1/(2*a),Int[(a+b*Coth[c+d*x])^(n+1),x]] /; 324FreeQ[{a,b,c,d},x] && ZeroQ[a^2-b^2] && RationalQ[n] && n<0 325 326 327(* ::PageBreak:: *) 328(**) 329 330 331(* ::Title::Bold::Closed:: *) 332(*\[Integral](a+b Tanh[c+d x])^n \[DifferentialD]x when a^2+b^2!=0*) 333 334 335(* ::Subsubsection:: *) 336(*Derivation: Algebraic expansion and integration by substitution*) 337 338 339(* ::Subsubsection:: *) 340(*Basis: 1/(a+b Tanh[z])=a/(a^2-b^2)-b/((a^2-b^2) (a Cosh[z]+b Sinh[z])) \!\( *) 341(*\*SubscriptBox[\(\[PartialD]\), \(z\)]\((a\ Cosh[z] + b\ Sinh[z])\)\)*) 342 343 344(* ::Subsubsection:: *) 345(*Rule: If a^2-b^2!=0, then*) 346 347 348(* ::Subsubtitle::Bold:: *) 349(*\[Integral]1/(a+b Tanh[c+d x]) \[DifferentialD]x \[LongRightArrow] ((a x)/(a^2-b^2))-(b Log[a Cosh[c+d x]+b Sinh[c+d x]])/(d (a^2-b^2))*) 350 351 352(* ::Subsubsection:: *) 353(*Program code:*) 354 355 356(* ::Code:: *) 357Int[1/(a_+b_.*Tanh[c_.+d_.*x_]),x_Symbol] := 358 a*x/(a^2-b^2) - b*Log[a*Cosh[c+d*x]+b*Sinh[c+d*x]]/(d*(a^2-b^2)) /; 359FreeQ[{a,b,c,d},x] && NonzeroQ[a^2-b^2] 360 361 362(* ::Code:: *) 363Int[1/(a_+b_.*Coth[c_.+d_.*x_]),x_Symbol] := 364 a*x/(a^2-b^2) - b*Log[b*Cosh[c+d*x]+a*Sinh[c+d*x]]/(d*(a^2-b^2)) /; 365FreeQ[{a,b,c,d},x] && NonzeroQ[a^2-b^2] 366 367 368(* ::PageBreak:: *) 369(**) 370 371 372(* ::Title::Bold::Closed:: *) 373(*\[Integral](A+B Tanh[c+d x])(a+b Tanh[c+d x])^n \[DifferentialD]x*) 374 375 376(* ::Subsubsection:: *) 377(*Derivation: Algebraic expansion*) 378 379 380(* ::Subsubsection:: *) 381(*Basis: (A+B z)/(a+b z)=B/b+(b A-a B)/b 1/(a+b z)*) 382 383 384(* ::Subsubsection:: *) 385(*Rule: If b A-a B!=0, then*) 386 387 388(* ::Subsubtitle::Bold:: *) 389(*\[Integral](A+B Tanh[c+d x])/(a+b Tanh[c+d x]) \[DifferentialD]x \[LongRightArrow] ((B x)/b)+(b A-a B)/b \[Integral]1/(a+b Tanh[c+d x]) \[DifferentialD]x*) 390 391 392(* ::Subsubsection:: *) 393(*Program code:*) 394 395 396(* ::Code:: *) 397Int[(A_.+B_.*Tanh[c_.+d_.*x_])/(a_.+b_.*Tanh[c_.+d_.*x_]),x_Symbol] := 398 B*x/b + Dist[(b*A-a*B)/b,Int[1/(a+b*Tanh[c+d*x]),x]] /; 399FreeQ[{a,b,c,d,A,B},x] && NonzeroQ[b*A-a*B] 400 401 402(* ::Code:: *) 403Int[(A_.+B_.*Coth[c_.+d_.*x_])/(a_.+b_.*Coth[c_.+d_.*x_]),x_Symbol] := 404 B*x/b + Dist[(b*A-a*B)/b,Int[1/(a+b*Coth[c+d*x]),x]] /; 405FreeQ[{a,b,c,d,A,B},x] && NonzeroQ[b*A-a*B] 406 407 408(* ::Subsubsection:: *) 409(**) 410 411 412(* ::Subsubsection:: *) 413(*Note: This rule does not appear in published integral tables.*) 414 415 416(* ::Subsubsection:: *) 417(*Rule: If A^2-B^2=0 \[And] b A+a B!=0, then*) 418 419 420(* ::Subsubtitle::Bold:: *) 421(*\[Integral](A+B Tanh[c+d x])/Sqrt[a+b Tanh[c+d x]] \[DifferentialD]x \[LongRightArrow] ((2 B)/(d Sqrt[(b A+a B)/B]))ArcTanh[Sqrt[a+b Tanh[c+d x]]/Sqrt[(b A+a B)/B]]*) 422 423 424(* ::Subsubsection:: *) 425(*Program code:*) 426 427 428(* ::Code:: *) 429Int[(A_+B_.*Tanh[c_.+d_.*x_])/Sqrt[a_.+b_.*Tanh[c_.+d_.*x_]],x_Symbol] := 430 2*B/(d*Sqrt[(b*A+a*B)/B])*ArcTanh[Sqrt[a+b*Tanh[c+d*x]]/Sqrt[(b*A+a*B)/B]] /; 431FreeQ[{a,b,c,d,A,B},x] && ZeroQ[A^2-B^2] && NonzeroQ[b*A+a*B] 432 433 434(* ::Code:: *) 435Int[(A_+B_.*Coth[c_.+d_.*x_])/Sqrt[a_.+b_.*Coth[c_.+d_.*x_]],x_Symbol] := 436 2*B/(d*Sqrt[(b*A+a*B)/B])*ArcTanh[Sqrt[a+b*Coth[c+d*x]]/Sqrt[(b*A+a*B)/B]] /; 437FreeQ[{a,b,c,d,A,B},x] && ZeroQ[A^2-B^2] && NonzeroQ[b*A+a*B] 438 439 440(* ::Subsubsection:: *) 441(**) 442 443 444(* ::Subsubsection:: *) 445(*Derivation: Algebraic expansion*) 446 447 448(* ::Subsubsection:: *) 449(*Basis: A+B z=(A+B)/2 (1+z)+(A-B)/2 (1-z)*) 450 451 452(* ::Subsubsection:: *) 453(*Rule: If A^2-B^2!=0 \[And] a^2-b^2!=0, then*) 454 455 456(* ::Subsubtitle::Bold:: *) 457(*\[Integral](A+B Tanh[c+d x])/Sqrt[a+b Tanh[c+d x]] \[DifferentialD]x \[LongRightArrow] ((A+B)/2)\[Integral](1+Tanh[c+d x])/Sqrt[a+b Tanh[c+d x]] \[DifferentialD]x+(A-B)/2 \[Integral](1-Tanh[c+d x])/Sqrt[a+b Tanh[c+d x]] \[DifferentialD]x*) 458 459 460(* ::Subsubsection:: *) 461(*Program code:*) 462 463 464(* ::Code:: *) 465Int[(A_.+B_.*Tanh[c_.+d_.*x_])/Sqrt[a_.+b_.*Tanh[c_.+d_.*x_]],x_Symbol] := 466 Dist[(A+B)/2,Int[(1+Tanh[c+d*x])/Sqrt[a+b*Tanh[c+d*x]],x]] + 467 Dist[(A-B)/2,Int[(1-Tanh[c+d*x])/Sqrt[a+b*Tanh[c+d*x]],x]] /; 468FreeQ[{a,b,c,d,A,B},x] && NonzeroQ[A^2-B^2] && NonzeroQ[a^2-b^2] 469 470 471(* ::Code:: *) 472Int[(A_.+B_.*Coth[c_.+d_.*x_])/Sqrt[a_.+b_.*Coth[c_.+d_.*x_]],x_Symbol] := 473 Dist[(A+B)/2,Int[(1+Coth[c+d*x])/Sqrt[a+b*Coth[c+d*x]],x]] + 474 Dist[(A-B)/2,Int[(1-Coth[c+d*x])/Sqrt[a+b*Coth[c+d*x]],x]] /; 475FreeQ[{a,b,c,d,A,B},x] && NonzeroQ[A^2-B^2] && NonzeroQ[a^2-b^2] 476 477 478(* ::Subsubsection:: *) 479(**) 480 481 482(* ::Subsubsection:: *) 483(*Note: This rule does not appear in published integral tables.*) 484 485 486(* ::Subsubsection:: *) 487(*Rule: If n>0, then*) 488 489 490(* ::Subsubtitle::Bold:: *) 491(*\[Integral](A+B Tanh[c+d x])(a+b Tanh[c+d x])^n \[DifferentialD]x \[LongRightArrow] *) 492(*-((B (a+b Tanh[c+d x])^n)/(d n))+\[Integral](a A+b B+(b A+a B) Tanh[c+d x])(a+b Tanh[c+d x])^(n-1) \[DifferentialD]x*) 493 494 495(* ::Subsubsection:: *) 496(*Program code:*) 497 498 499(* ::Code:: *) 500Int[(A_.+B_.*Tanh[c_.+d_.*x_])*(a_+b_.*Tanh[c_.+d_.*x_])^n_.,x_Symbol] := 501 -B*(a+b*Tanh[c+d*x])^n/(d*n) + 502 Int[(a*A+b*B+(b*A+a*B)*Tanh[c+d*x])*(a+b*Tanh[c+d*x])^(n-1),x] /; 503FreeQ[{a,b,c,d,A,B},x] && RationalQ[n] && n>0 504 505 506(* ::Code:: *) 507Int[(A_.+B_.*Coth[c_.+d_.*x_])*(a_+b_.*Coth[c_.+d_.*x_])^n_.,x_Symbol] := 508 -B*(a+b*Coth[c+d*x])^n/(d*n) + 509 Int[(a*A+b*B+(b*A+a*B)*Coth[c+d*x])*(a+b*Coth[c+d*x])^(n-1),x] /; 510FreeQ[{a,b,c,d,A,B},x] && RationalQ[n] && n>0 511 512 513(* ::Subsubsection:: *) 514(**) 515 516 517(* ::Subsubsection:: *) 518(*Note: This rule does not appear in published integral tables.*) 519 520 521(* ::Subsubsection:: *) 522(*Rule: If a^2-b^2!=0 \[And] n<-1, then*) 523 524 525(* ::Subsubtitle::Bold:: *) 526(*\[Integral](A+B Tanh[c+d x])(a+b Tanh[c+d x])^n \[DifferentialD]x \[LongRightArrow] *) 527(*-(((b A-a B) (a+b Tanh[c+d x])^(n+1))/(d(a^2-b^2)(n+1)))+1/(a^2-b^2) \[Integral](a A-b B-(b A-a B) Tanh[c+d x])(a+b Tanh[c+d x])^(n+1) \[DifferentialD]x*) 528 529 530(* ::Subsubsection:: *) 531(*Program code:*) 532 533 534(* ::Code:: *) 535Int[(A_.+B_.*Tanh[c_.+d_.*x_])*(a_+b_.*Tanh[c_.+d_.*x_])^n_,x_Symbol] := 536 -(b*A-a*B)*(a+b*Tanh[c+d*x])^(n+1)/(d*(a^2-b^2)*(n+1)) + 537 Dist[1/(a^2-b^2),Int[(a*A-b*B-(b*A-a*B)*Tanh[c+d*x])*(a+b*Tanh[c+d*x])^(n+1),x]] /; 538FreeQ[{a,b,c,d,A,B},x] && NonzeroQ[a^2-b^2] && RationalQ[n] && n<-1 539 540 541(* ::Code:: *) 542Int[(A_.+B_.*Coth[c_.+d_.*x_])*(a_+b_.*Coth[c_.+d_.*x_])^n_,x_Symbol] := 543 -(b*A-a*B)*(a+b*Coth[c+d*x])^(n+1)/(d*(a^2-b^2)*(n+1)) + 544 Dist[1/(a^2-b^2),Int[(a*A-b*B-(b*A-a*B)*Coth[c+d*x])*(a+b*Coth[c+d*x])^(n+1),x]] /; 545FreeQ[{a,b,c,d,A,B},x] && NonzeroQ[a^2-b^2] && RationalQ[n] && n<-1 546 547 548(* ::PageBreak:: *) 549(**) 550 551 552(* ::Title::Bold::Closed:: *) 553(*\[Integral](a+b Tan[c+d x]^2)^n \[DifferentialD]x*) 554 555 556(* ::Subsubsection:: *) 557(*Derivation: Algebraic simplification*) 558 559 560(* ::Subsubsection:: *) 561(*Basis: If a-b=0, then a+b Tan[z]^2=b Sec[z]^2*) 562 563 564(* ::Subsubsection:: *) 565(*Rule: If a-b=0 \[And] m\[Element]\[DoubleStruckCapitalZ], then*) 566 567 568(* ::Subsubtitle::Bold:: *) 569(*\[Integral]u (a+b Tan[v]^2)^m \[DifferentialD]x \[LongRightArrow] b^m\[Integral]u Sec[v]^(2m) \[DifferentialD]x*) 570 571 572(* ::Subsubsection:: *) 573(*Program code:*) 574 575 576(* ::Code:: *) 577Int[u_.*(a_+b_.*Tan[v_]^2)^m_,x_Symbol] := 578 Dist[b^m,Int[u*Sec[v]^(2*m),x]] /; 579FreeQ[{a,b,m},x] && ZeroQ[a-b] && IntegerQ[m] 580 581 582(* ::Subsubsection:: *) 583(**) 584 585 586(* ::Subsubsection:: *) 587(*Derivation: Algebraic simplification*) 588 589 590(* ::Subsubsection:: *) 591(*Basis: If a-b=0, then a+b Tan[z]^2=b Sec[z]^2*) 592 593 594(* ::Subsubsection:: *) 595(*Rule: If a-b=0 \[And] m\[NotElement]\[DoubleStruckCapitalZ], then*) 596 597 598(* ::Subsubtitle::Bold:: *) 599(*\[Integral]u (a+b Tan[v]^2)^m \[DifferentialD]x \[LongRightArrow] \[Integral]u (b Sec[v]^2)^m \[DifferentialD]x*) 600 601 602(* ::Subsubsection:: *) 603(*Program code:*) 604 605 606(* ::Code:: *) 607Int[u_.*(a_+b_.*Tan[v_]^2)^m_,x_Symbol] := 608 Int[u*(b*Sec[v]^2)^m,x] /; 609FreeQ[{a,b,m},x] && ZeroQ[a-b] && Not[IntegerQ[m]] 610 611 612(* ::Subsubsection:: *) 613(**) 614 615 616(* ::Subsubsection:: *) 617(*Rule: If a+b!=0, then*) 618 619 620(* ::Subsubtitle::Bold:: *) 621(*\[Integral]1/(a+b Tanh[c+d x]^2) \[DifferentialD]x \[LongRightArrow] (x/(a+b))+Sqrt[b]/(Sqrt[a] d (a+b)) ArcTan[(Sqrt[b] Tanh[c+d x])/Sqrt[a]]*) 622 623 624(* ::Subsubsection:: *) 625(*Program code:*) 626 627 628(* ::Code:: *) 629Int[1/(a_+b_.*Tanh[c_.+d_.*x_]^2),x_Symbol] := 630 x/(a+b) + Sqrt[b]*ArcTan[Sqrt[b]*Tanh[c+d*x]/Sqrt[a]]/(Sqrt[a]*d*(a+b)) /; 631FreeQ[{a,b,c,d},x] && NonzeroQ[a+b] 632 633 634(* ::Code:: *) 635Int[1/(a_+b_.*Coth[c_.+d_.*x_]^2),x_Symbol] := 636 x/(a+b) + Sqrt[b]*ArcTan[Sqrt[b]*Coth[c+d*x]/Sqrt[a]]/(Sqrt[a]*d*(a+b)) /; 637FreeQ[{a,b,c,d},x] && NonzeroQ[a+b] 638 639 640(* ::PageBreak:: *) 641(**) 642 643 644(* ::Title::Bold::Closed:: *) 645(*\[Integral]x^m Tanh[a+b x^n]^p \[DifferentialD]x*) 646 647 648(* ::Subsubsection:: *) 649(*Derivation: Algebraic expansion*) 650 651 652(* ::Subsubsection:: *) 653(*Basis: Tanh[z]=1-2/(1+E^(2z))*) 654 655 656(* ::Subsubsection:: *) 657(*Rule: If m\[Element]\[DoubleStruckCapitalZ] \[And] m>0 \[And] m-n+1!=0, then*) 658 659 660(* ::Subsubtitle::Bold:: *) 661(*\[Integral]x^m Tanh[a+b x^n]\[DifferentialD]x \[LongRightArrow] (x^(m+1)/(m+1))-2\[Integral]x^m/(1+E^(2a+2b x^n)) \[DifferentialD]x*) 662 663 664(* ::Subsubsection:: *) 665(*Program code:*) 666 667 668(* ::Code:: *) 669Int[x_^m_.*Tanh[a_.+b_.*x_^n_.],x_Symbol] := 670 x^(m+1)/(m+1) - 671 Dist[2,Int[x^m/(1+E^(2*a+2*b*x^n)),x]] /; 672FreeQ[{a,b,n},x] && IntegerQ[m] && m>0 && NonzeroQ[m-n+1] 673 674 675(* ::Code:: *) 676Int[x_^m_.*Coth[a_.+b_.*x_^n_.],x_Symbol] := 677 x^(m+1)/(m+1) - 678 Dist[2,Int[x^m/(1-E^(2*a+2*b*x^n)),x]] /; 679FreeQ[{a,b,n},x] && IntegerQ[m] && m>0 && NonzeroQ[m-n+1] 680 681 682(* ::Subsubsection:: *) 683(**) 684 685 686(* ::Subsubsection:: *) 687(*Note: This rule does not appear in published integral tables.*) 688 689 690(* ::Subsubsection:: *) 691(*Rule: If p>1 \[And] m-n+1!=0 \[And] 0<n<=m, then*) 692 693 694(* ::Subsubtitle::Bold:: *) 695(*\[Integral]x^m Tanh[a+b x^n]^p \[DifferentialD]x \[LongRightArrow] -((x^(m-n+1) Tanh[a+b x^n]^(p-1))/(b n (p-1)))+*) 696(*(m-n+1)/(b n (p-1)) \[Integral]x^(m-n) Tanh[a+b x^n]^(p-1) \[DifferentialD]x+\[Integral]x^m Tanh[a+b x^n]^(p-2) \[DifferentialD]x*) 697 698 699(* ::Subsubsection:: *) 700(*Program code:*) 701 702 703(* ::Code:: *) 704Int[x_^m_.*Tanh[a_.+b_.*x_^n_.]^p_,x_Symbol] := 705 -x^(m-n+1)*Tanh[a+b*x^n]^(p-1)/(b*n*(p-1)) + 706 Dist[(m-n+1)/(b*n*(p-1)),Int[x^(m-n)*Tanh[a+b*x^n]^(p-1),x]] + 707 Int[x^m*Tanh[a+b*x^n]^(p-2),x] /; 708FreeQ[{a,b},x] && RationalQ[{m,n,p}] && p>1 && NonzeroQ[m-n+1] && 0<n<=m 709 710 711(* ::Code:: *) 712Int[x_^m_.*Coth[a_.+b_.*x_^n_.]^p_,x_Symbol] := 713 -x^(m-n+1)*Coth[a+b*x^n]^(p-1)/(b*n*(p-1)) + 714 Dist[(m-n+1)/(b*n*(p-1)),Int[x^(m-n)*Coth[a+b*x^n]^(p-1),x]] + 715 Int[x^m*Coth[a+b*x^n]^(p-2),x] /; 716FreeQ[{a,b},x] && RationalQ[{m,n,p}] && p>1 && NonzeroQ[m-n+1] && 0<n<=m 717 718 719(* ::PageBreak:: *) 720(**) 721 722 723(* ::Title::Bold::Closed:: *) 724(*\[Integral]x^m Tanh[a+b x+c x^2]\[DifferentialD]x*) 725 726 727(* ::Subsubsection:: *) 728(*Rule:*) 729 730 731(* ::Subsubtitle::Bold:: *) 732(*\[Integral]x Tanh[a+b x+c x^2]\[DifferentialD]x \[LongRightArrow] (Log[Cosh[a+b x+c x^2]]/(2 c))-b/(2 c) \[Integral]Tanh[a+b x+c x^2]\[DifferentialD]x*) 733 734 735(* ::Subsubsection:: *) 736(*Program code:*) 737 738 739(* ::Code:: *) 740Int[x_*Tanh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := 741 Log[Cosh[a+b*x+c*x^2]]/(2*c) - 742 Dist[b/(2*c),Int[Tanh[a+b*x+c*x^2],x]] /; 743FreeQ[{a,b,c},x] 744 745 746(* ::Code:: *) 747Int[x_*Coth[a_.+b_.*x_+c_.*x_^2],x_Symbol] := 748 Log[Sinh[a+b*x+c*x^2]]/(2*c) - 749 Dist[b/(2*c),Int[Coth[a+b*x+c*x^2],x]] /; 750FreeQ[{a,b,c},x] 751 752 753(* ::Subsubsection:: *) 754(**) 755 756 757(* ::Subsubsection:: *) 758(*Note: This rule is valid, but to be useful need a rule for reducing integrands of the form x^m Log[Cosh[a+b x+c x^2]].*) 759 760 761(* ::Subsubsection:: *) 762(*Rule: If m>1, then*) 763 764 765(* ::Subsubtitle::Bold:: *) 766(*\[Integral]x^m Tanh[a+b x+c x^2]\[DifferentialD]x \[LongRightArrow] ((x^(m-1) Log[Cosh[a+b x+c x^2]])/(2 c))-*) 767(*b/(2 c) \[Integral]x^(m-1) Tanh[a+b x+c x^2]\[DifferentialD]x-(m-1)/(2 c) \[Integral]x^(m-2) Log[Cosh[a+b x+c x^2]]\[DifferentialD]x*) 768 769 770(* ::Subsubsection:: *) 771(*Program code:*) 772 773 774(* ::Code:: *) 775(* Int[x_^m_*Tanh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := 776 x^(m-1)*Log[Cosh[a+b*x+c*x^2]]/(2*c) - 777 Dist[b/(2*c),Int[x^(m-1)*Tanh[a+b*x+c*x^2],x]] - 778 Dist[(m-1)/(2*c),Int[x^(m-2)*Log[Cosh[a+b*x+c*x^2]],x]] /; 779FreeQ[{a,b,c},x] && RationalQ[m] && m>1 *) 780 781 782(* ::Code:: *) 783(* Int[x_^m_*Coth[a_.+b_.*x_+c_.*x_^2],x_Symbol] := 784 x^(m-1)*Log[Sinh[a+b*x+c*x^2]]/(2*c) - 785 Dist[b/(2*c),Int[x^(m-1)*Coth[a+b*x+c*x^2],x]] - 786 Dist[(m-1)/(2*c),Int[x^(m-2)*Log[Sinh[a+b*x+c*x^2]],x]] /; 787FreeQ[{a,b,c},x] && RationalQ[m] && m>1 *) 788 789 790(* ::PageBreak:: *) 791(**) 792