Optimal. Leaf size=87 \[ \frac {F_1\left (\frac {1}{2};2,-p;\frac {3}{2};\cosh ^2(e+f x),-\frac {b \cosh ^2(e+f x)}{a-b}\right ) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^p \left (1+\frac {b \cosh ^2(e+f x)}{a-b}\right )^{-p}}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 441, 440}
\begin {gather*} \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} F_1\left (\frac {1}{2};2,-p;\frac {3}{2};\cosh ^2(e+f x),-\frac {b \cosh ^2(e+f x)}{a-b}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 440
Rule 441
Rule 3265
Rubi steps
\begin {align*} \int \text {csch}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^p}{\left (1-x^2\right )^2} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac {\left (\left (a-b+b \cosh ^2(e+f x)\right )^p \left (1+\frac {b \cosh ^2(e+f x)}{a-b}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a-b}\right )^p}{\left (1-x^2\right )^2} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {1}{2};2,-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}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 81.55, size = 0, normalized size = 0.00 \begin {gather*} \int \text {csch}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.27, size = 0, normalized size = 0.00 \[\int \mathrm {csch}\left (f x +e \right )^{3} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.45, size = 25, normalized size = 0.29 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname {csch}\left (f x + e\right )^{3}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p}{{\mathrm {sinh}\left (e+f\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________