Integrand size = 21, antiderivative size = 88 \[ \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},\cosh ^2(e+f x),-\frac {b \cosh ^2(e+f x)}{a-b}\right ) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^p \left (1+\frac {b \cosh ^2(e+f x)}{a-b}\right )^{-p}}{f} \]
-AppellF1(1/2,1,-p,3/2,cosh(f*x+e)^2,-b*cosh(f*x+e)^2/(a-b))*cosh(f*x+e)*( a-b+b*cosh(f*x+e)^2)^p/f/((1+b*cosh(f*x+e)^2/(a-b))^p)
\[ \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 3665, 334, 333}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \left (a-b \sin (i e+i f x)^2\right )^p}{\sin (i e+i f x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\left (a-b \sin (i e+i f x)^2\right )^p}{\sin (i e+i f x)}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \frac {\left (b \cosh ^2(e+f x)+a-b\right )^p}{1-\cosh ^2(e+f x)}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 334 |
\(\displaystyle -\frac {\left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} \int \frac {\left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^p}{1-\cosh ^2(e+f x)}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle -\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},\cosh ^2(e+f x),-\frac {b \cosh ^2(e+f x)}{a-b}\right )}{f}\) |
-((AppellF1[1/2, 1, -p, 3/2, Cosh[e + f*x]^2, -((b*Cosh[e + f*x]^2)/(a - b ))]*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p)/(f*(1 + (b*Cosh[e + f*x]^ 2)/(a - b))^p))
3.2.34.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[ (1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && !(IntegerQ[p] || GtQ[a, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
\[\int \operatorname {csch}\left (f x +e \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname {csch}\left (f x + e\right ) \,d x } \]
Timed out. \[ \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
\[ \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname {csch}\left (f x + e\right ) \,d x } \]
\[ \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname {csch}\left (f x + e\right ) \,d x } \]
Timed out. \[ \int \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p}{\mathrm {sinh}\left (e+f\,x\right )} \,d x \]