Lines Matching refs:Sinh
4 (*\[Integral](x (A+B Sinh[c+d x]))/(a+b Sinh[c+d x])^2 \[DifferentialD]x*)
16 …Sinh[c+d x]))/(a+b Sinh[c+d x])^2 \[DifferentialD]x \[LongRightArrow] ((B x Cosh[c+d x])/(a d (a…
24 Int[x_*(A_+B_.*Sinh[c_.+d_.*x_])/(a_+b_.*Sinh[c_.+d_.*x_])^2,x_Symbol] :=
25 B*x*Cosh[c+d*x]/(a*d*(a+b*Sinh[c+d*x])) -
26 Dist[B/(a*d),Int[Cosh[c+d*x]/(a+b*Sinh[c+d*x]),x]] /;
32 B*x*Sinh[c+d*x]/(a*d*(a+b*Cosh[c+d*x])) -
33 Dist[B/(a*d),Int[Sinh[c+d*x]/(a+b*Cosh[c+d*x]),x]] /;
42 (*\[Integral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x*)
54 (*Basis: Sinh[z] Tanh[z]=Cosh[z]-Sech[z]*)
62 (*\[Integral]Sinh[a+b x] Tanh[a+b x]\[DifferentialD]x \[LongRightArrow] (Sinh[a+b x]/b)-\[Integra…
70 Int[Sinh[a_.+b_.*x_]*Tanh[a_.+b_.*x_],x_Symbol] :=
71 Sinh[a+b*x]/b - Int[Sech[a+b*x],x] /;
94 (*\[Integral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^m Tanh…
102 Int[Sinh[a_.+b_.*x_]^m_*Tanh[a_.+b_.*x_]^n_,x_Symbol]:=
103 Sinh[a+b*x]^m*Tanh[a+b*x]^(n-1)/(b*m) /;
122 (*Basis: If m,n,(m+n-1)/2\[Element]\[DoubleStruckCapitalZ], then Sinh[z]^m Tanh[z]^n=(-1+Cosh[z]^2…
135 (*\[Integral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] (1/b)Subst[\[Integra…
143 Int[Sinh[a_.+b_.*x_]^m_.*Tanh[a_.+b_.*x_]^n_.,x_Symbol] :=
149 …\[Element]\[DoubleStruckCapitalZ], then Cosh[z]^m Coth[z]^n=(1+Sinh[z]^2)^((m+n-1)/2)/Sinh[z]^n \!…
150 (*\*SubscriptBox[\(\[PartialD]\), \(z\)]\(Sinh[z]\)\)*)
155 Dist[1/b,Subst[Int[Regularize[(1+x^2)^((m+n-1)/2)/x^n,x],x],x,Sinh[a+b*x]]] /;
172 …\[Integral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^m Tanh[…
180 Int[Sinh[a_.+b_.*x_]^m_*Tanh[a_.+b_.*x_]^n_,x_Symbol] :=
181 Sinh[a+b*x]^m*Tanh[a+b*x]^(n+1)/(b*m) -
182 Dist[(n+1)/m,Int[Sinh[a+b*x]^(m-2)*Tanh[a+b*x]^(n+2),x]] /;
210 …gral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^(m+2) Tanh[a+…
218 Int[Sinh[a_.+b_.*x_]^m_*Tanh[a_.+b_.*x_]^n_,x_Symbol] :=
219 Sinh[a+b*x]^(m+2)*Tanh[a+b*x]^(n-1)/(b*(n-1)) -
220 Dist[(m+2)/(n-1),Int[Sinh[a+b*x]^(m+2)*Tanh[a+b*x]^(n-2),x]] /;
248 …[Integral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^m Tanh[a…
256 Int[Sinh[a_.+b_.*x_]^m_*Tanh[a_.+b_.*x_]^n_.,x_Symbol]:=
257 Sinh[a+b*x]^m*Tanh[a+b*x]^(n-1)/(b*m) -
258 Dist[(m+n-1)/m,Int[Sinh[a+b*x]^(m-2)*Tanh[a+b*x]^n,x]] /;
286 …gral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] -((Sinh[a+b x]^m Tanh[a+b x…
294 Int[Sinh[a_.+b_.*x_]^m_.*Tanh[a_.+b_.*x_]^n_,x_Symbol] :=
295 -Sinh[a+b*x]^m*Tanh[a+b*x]^(n-1)/(b*(n-1)) +
296 Dist[(m+n-1)/(n-1),Int[Sinh[a+b*x]^m*Tanh[a+b*x]^(n-2),x]] /;
324 …al]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^(m+2) Tanh[a+b …
332 Int[Sinh[a_.+b_.*x_]^m_*Tanh[a_.+b_.*x_]^n_.,x_Symbol]:=
333 Sinh[a+b*x]^(m+2)*Tanh[a+b*x]^(n-1)/(b*(m+n+1)) -
334 Dist[(m+2)/(m+n+1),Int[Sinh[a+b*x]^(m+2)*Tanh[a+b*x]^n,x]] /;
362 …gral]Sinh[a+b x]^m Tanh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^m Tanh[a+b x]…
370 Int[Sinh[a_.+b_.*x_]^m_.*Tanh[a_.+b_.*x_]^n_,x_Symbol]:=
371 Sinh[a+b*x]^m*Tanh[a+b*x]^(n+1)/(b*(m+n+1)) +
372 Dist[(n+1)/(m+n+1),Int[Sinh[a+b*x]^m*Tanh[a+b*x]^(n+2),x]] /;
392 (*\[Integral]u Sinh[v] Hyper[w]\[DifferentialD]x*)
400 (*Basis: Sinh[v] Cosh[w]=1/2 Sinh[v+w]+1/2 Sinh[v-w]*)
408 (*\[Integral]u Sinh[v] Cosh[w]\[DifferentialD]x \[LongRightArrow] (1/2)\[Integral]u Sinh[v+w]\[Di…
416 Int[u_.*Sinh[v_]*Cosh[w_],x_Symbol] :=
417 Dist[1/2,Int[u*Regularize[Sinh[v+w],x],x]] +
418 Dist[1/2,Int[u*Regularize[Sinh[v-w],x],x]] /;
431 (*Basis: Sinh[v] Sinh[w]=1/2 Cosh[v+w]-1/2 Cosh[v-w]*)
439 (*\[Integral]u Sinh[v] Sinh[w]\[DifferentialD]x \[LongRightArrow] (1/2)\[Integral]u Cosh[v+w]\[Di…
447 Int[u_.*Sinh[v_]*Sinh[w_],x_Symbol] :=
473 (*Basis: Sinh[v] Tanh[w]=Cosh[v]-Cosh[v-w] Sech[w]*)
481 (*\[Integral]u Sinh[v] Tanh[w]^n \[DifferentialD]x \[LongRightArrow] \[Integral]u Cosh[v] Tanh[w]…
489 Int[u_.*Sinh[v_]*Tanh[w_]^n_.,x_Symbol] :=
495 (*Basis: Cosh[v] Coth[w]=Sinh[v]+Cosh[v-w] Csch[w]*)
500 Int[u*Sinh[v]*Coth[w]^(n-1),x] + Cosh[v-w]*Int[u*Csch[w]*Coth[w]^(n-1),x] /;
513 (*Basis: Sinh[v] Coth[w]=Cosh[v]+Sinh[v-w] Csch[w]*)
521 (*\[Integral]u Sinh[v] Coth[w]^n \[DifferentialD]x \[LongRightArrow] \[Integral]u Cosh[v] Coth[w]…
529 Int[u_.*Sinh[v_]*Coth[w_]^n_.,x_Symbol] :=
530 Int[u*Cosh[v]*Coth[w]^(n-1),x] + Sinh[v-w]*Int[u*Csch[w]*Coth[w]^(n-1),x] /;
535 (*Basis: Cosh[v] Tanh[w]=Sinh[v]-Sinh[v-w] Sech[w]*)
540 Int[u*Sinh[v]*Tanh[w]^(n-1),x] - Sinh[v-w]*Int[u*Sech[w]*Tanh[w]^(n-1),x] /;
553 (*Basis: Sinh[v] Sech[w]=Cosh[v-w] Tanh[w]+Sinh[v-w]*)
561 (*\[Integral]u Sinh[v] Sech[w]^n \[DifferentialD]x \[LongRightArrow] Cosh[v-w] \[Integral]u Tanh[…
569 Int[u_.*Sinh[v_]*Sech[w_]^n_.,x_Symbol] :=
570 Cosh[v-w]*Int[u*Tanh[w]*Sech[w]^(n-1),x] + Sinh[v-w]*Int[u*Sech[w]^(n-1),x] /;
575 (*Basis: Cosh[v]*Csch[w]=Cosh[v-w]*Coth[w]+Sinh[v-w]*)
580 Cosh[v-w]*Int[u*Coth[w]*Csch[w]^(n-1),x] + Sinh[v-w]*Int[u*Csch[w]^(n-1),x] /;
593 (*Basis: Sinh[v] Csch[w]=Sinh[v-w] Coth[w]+Cosh[v-w]*)
601 (*\[Integral]u Sinh[v] Csch[w]^n \[DifferentialD]x \[LongRightArrow] Sinh[v-w] \[Integral]u Coth[…
609 Int[u_.*Sinh[v_]*Csch[w_]^n_.,x_Symbol] :=
610 Sinh[v-w]*Int[u*Coth[w]*Csch[w]^(n-1),x] + Cosh[v-w]*Int[u*Csch[w]^(n-1),x] /;
615 (*Basis: Cosh[v] Sech[w]=Sinh[v-w] Tanh[w]+Cosh[v-w]*)
620 Sinh[v-w]*Int[u*Tanh[w]*Sech[w]^(n-1),x] + Cosh[v-w]*Int[u*Sech[w]^(n-1),x] /;
629 (*\[Integral]x^m Sinh[a+b x^n]^p Cosh[a+b x^n]\[DifferentialD]x*)
641 …Sinh[a+b x^n]^p Cosh[a+b x^n]\[DifferentialD]x \[LongRightArrow] ((x^(m-n+1) Sinh[a+b x^n]^(p+1)…
649 Int[x_^m_.*Sinh[a_.+b_.*x_^n_.]^p_.*Cosh[a_.+b_.*x_^n_.],x_Symbol] :=
650 x^(m-n+1)*Sinh[a+b*x^n]^(p+1)/(b*n*(p+1)) -
651 Dist[(m-n+1)/(b*n*(p+1)),Int[x^(m-n)*Sinh[a+b*x^n]^(p+1),x]] /;
660 Int[x_^m_.*Cosh[a_.+b_.*x_^n_.]^p_.*Sinh[a_.+b_.*x_^n_.],x_Symbol] :=
675 (*\[Integral]Sinh[a+b x]^m Cosh[a+b x]^n \[DifferentialD]x*)
687 (*\[Integral]Sinh[a+b x]^m Cosh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^(m+1) …
695 Int[Sinh[a_.+b_.*x_]^m_.*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
696 Sinh[a+b*x]^(m+1)*Cosh[a+b*x]^(n+1)/(b*(m+1)) /;
709 …\[Element]\[DoubleStruckCapitalZ], then Sinh[z]^m Cosh[z]^n=Sinh[z]^m (1+Sinh[z]^2)^((n-1)/2) Deri…
721 …Integral]Sinh[a+b x]^m Cosh[a+b x]^n \[DifferentialD]x \[LongRightArrow] (1/b)Subst[\[Integral]x…
729 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
730 Dist[1/b,Subst[Int[Regularize[x^m*(1+x^2)^((n-1)/2),x],x],x,Sinh[a+b*x]]] /;
735 (*Basis: If (m-1)/2\[Element]\[DoubleStruckCapitalZ], then Sinh[z]^m Cosh[z]^n=Cosh[z]^n (-1+Cosh[z…
739 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
757 …gral]Sinh[a+b x]^m Cosh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^(m-1) Cosh[a+…
765 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
766 Sinh[a+b*x]^(m-1)*Cosh[a+b*x]^(n+1)/(b*(n+1)) -
767 Dist[(m-1)/(n+1),Int[Sinh[a+b*x]^(m-2)*Cosh[a+b*x]^(n+2),x]] /;
776 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
777 Sinh[a+b*x]^(m+1)*Cosh[a+b*x]^(n-1)/(b*(m+1)) -
778 Dist[(n-1)/(m+1),Int[Sinh[a+b*x]^(m+2)*Cosh[a+b*x]^(n-2),x]] /;
795 …gral]Sinh[a+b x]^m Cosh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^(m-1) Cosh[a+…
803 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
804 Sinh[a+b*x]^(m-1)*Cosh[a+b*x]^(n+1)/(b*(m+n)) -
805 Dist[(m-1)/(m+n),Int[Sinh[a+b*x]^(m-2)*Cosh[a+b*x]^n,x]] /;
814 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
815 Sinh[a+b*x]^(m+1)*Cosh[a+b*x]^(n-1)/(b*(m+n)) +
816 Dist[(n-1)/(m+n),Int[Sinh[a+b*x]^m*Cosh[a+b*x]^(n-2),x]] /;
833 …ral]Sinh[a+b x]^m Cosh[a+b x]^n \[DifferentialD]x \[LongRightArrow] ((Sinh[a+b x]^(m+1) Cosh[a+b…
841 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
842 Sinh[a+b*x]^(m+1)*Cosh[a+b*x]^(n+1)/(b*(m+1)) -
843 Dist[(m+n+2)/(m+1),Int[Sinh[a+b*x]^(m+2)*Cosh[a+b*x]^n,x]] /;
852 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
853 -Sinh[a+b*x]^(m+1)*Cosh[a+b*x]^(n+1)/(b*(n+1)) +
854 Dist[(m+n+2)/(n+1),Int[Sinh[a+b*x]^m*Cosh[a+b*x]^(n+2),x]] /;
867 … If 1/m\[Element]\[DoubleStruckCapitalZ], then Sinh[z]^m/Cosh[z]^m=(Sinh[z]^m/Cosh[z]^m)^(1/m)/(m …
870 (*SuperscriptBox[\(Sinh[z]\), \(m\)], *)
883 …ntegral]Sinh[a+b x]^m/Cosh[a+b x]^m \[DifferentialD]x \[LongRightArrow] (1/(b m))Subst[\[Integra…
891 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
892 Dist[1/(b*m),Subst[Int[x^(1/m)/(1-x^(2/m)),x],x,Sinh[a+b*x]^m/Cosh[a+b*x]^m]] /;
897 …lement]\[DoubleStruckCapitalZ], then Cosh[z]^n/Sinh[z]^n=(Cosh[z]^n/Sinh[z]^n)^(1/n)/(n (1-(Cosh[z…
901 (*SuperscriptBox[\(Sinh[z]\), \(n\)]]\)*)
905 Int[Sinh[a_.+b_.*x_]^m_*Cosh[a_.+b_.*x_]^n_,x_Symbol] :=
906 Dist[1/(b*n),Subst[Int[x^(1/n)/(1-x^(2/n)),x],x,Cosh[a+b*x]^n/Sinh[a+b*x]^n]] /;
915 (*\[Integral]x^m (a+b Cosh[d+e x]^2+c Sinh[d+e x]^2)^n \[DifferentialD]x*)
923 (*Basis: a+b Cosh[z]^2+c Sinh[z]^2=1/2 (2 a+b-c+(b+c) Cosh[2 z])*)
931 (*\[Integral]x^m/(a+b Cosh[d+e x]^2+c Sinh[d+e x]^2) \[DifferentialD]x \[LongRightArrow] 2\[Integ…
939 Int[x_^m_./(a_.+b_.*Cosh[d_.+e_.*x_]^2+c_.*Sinh[d_.+e_.*x_]^2),x_Symbol] :=
949 (*\[Integral]x^m (a+b Sinh[c+d x] Cosh[c+d x])^n \[DifferentialD]x*)
957 (*Basis: Sinh[z] Cosh[z]=1/2 Sinh[2 z]*)
965 (*\[Integral]x^m/(a+b Sinh[c+d x] Cosh[c+d x]) \[DifferentialD]x \[LongRightArrow] \[Integral]x^m…
973 Int[x_^m_./(a_+b_.*Sinh[c_.+d_.*x_]*Cosh[c_.+d_.*x_]),x_Symbol] :=
974 Int[x^m/(a+b*Sinh[2*c+2*d*x]/2),x] /;
987 (*Basis: Sinh[z] Cosh[z]=1/2 Sinh[2 z]*)
995 (*\[Integral](a+b Sinh[c+d x] Cosh[c+d x])^n \[DifferentialD]x \[LongRightArrow] \[Integral](a+1/…
1003 Int[(a_+b_.*Sinh[c_.+d_.*x_]*Cosh[c_.+d_.*x_])^n_,x_Symbol] :=
1004 Int[(a+b*Sinh[2*c+2*d*x]/2)^n,x] /;
1013 (*\[Integral]Sinh[a+b x^n]^p Cosh[a+b x^n]^p \[DifferentialD]x*)
1021 (*Basis: Sinh[z] Cosh[z]=1/2 Sinh[2 z]*)
1029 (*\[Integral]Sinh[a+b x^n]^p Cosh[a+b x^n]^p \[DifferentialD]x \[LongRightArrow] (1/2)\[Integral]…
1037 Int[Sinh[a_.+b_.*x_^n_]^p_.*Cosh[a_.+b_.*x_^n_]^p_.,x_Symbol] :=
1038 Dist[1/2,Int[Sinh[2*a+2*b*x^n]^p,x]] /;
1051 (*\[Integral](a Csch[c+d x]+a Sinh[c+d x])^n \[DifferentialD]x*)
1059 (*Basis: Csch[z]+Sinh[z]=Cosh[z] Coth[z]*)
1067 (*\[Integral](a Csch[c+d x]+a Sinh[c+d x])^n \[DifferentialD]x \[LongRightArrow] \[Integral](a Co…
1075 Int[(a_.*Csch[c_.+d_.*x_]+b_.*Sinh[c_.+d_.*x_])^n_,x_Symbol] :=
1082 Int[(-a*Sinh[c+d*x]*Tanh[c+d*x])^n,x] /;
1099 (*Basis: (a+b Tanh[z])/Sech[z]=a Cosh[z]+b Sinh[z]*)
1107 … Tanh[v])^n \[DifferentialD]x \[LongRightArrow] \[Integral](a Cosh[v]+b Sinh[v])^n \[Differentia…
1116 Int[(a*Cosh[v]+b*Sinh[v])^n,x] /;
1122 Int[(b*Cosh[v]+a*Sinh[v])^n,x] /;
1241 (*Basis: a Sech[z]+b Tanh[z]=(a+b Sinh[z])/Cosh[z]*)
1249 …m+b Sech[v]^m)^n \[DifferentialD]x \[LongRightArrow] \[Integral](u (a+b Sinh[v]^m)^n)/Cosh[v]^(m…
1258 Int[u*(a+b*Sinh[v]^m)^n/Cosh[v]^(m*n),x] /;
1264 Int[u*(a+b*Cosh[v]^m)^n/Sinh[v]^(m*n),x] /;