Optimal. Leaf size=103 \[ -\frac{\sqrt{\cosh ^2(e+f x)} \text{csch}^3(e+f x) \text{sech}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac{b \sinh ^2(e+f x)}{a}+1\right )^{-p} F_1\left (-\frac{3}{2};\frac{1}{2},-p;-\frac{1}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right )}{3 f} \]
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Rubi [A] time = 0.104743, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3188, 511, 510} \[ -\frac{\sqrt{\cosh ^2(e+f x)} \text{csch}^3(e+f x) \text{sech}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac{b \sinh ^2(e+f x)}{a}+1\right )^{-p} F_1\left (-\frac{3}{2};\frac{1}{2},-p;-\frac{1}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 3188
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx &=\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^p}{x^4 \sqrt{1+x^2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac{b \sinh ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b x^2}{a}\right )^p}{x^4 \sqrt{1+x^2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac{F_1\left (-\frac{3}{2};\frac{1}{2},-p;-\frac{1}{2};-\sinh ^2(e+f x),-\frac{b \sinh ^2(e+f x)}{a}\right ) \sqrt{\cosh ^2(e+f x)} \text{csch}^3(e+f x) \text{sech}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac{b \sinh ^2(e+f x)}{a}\right )^{-p}}{3 f}\\ \end{align*}
Mathematica [F] time = 10.5505, size = 0, normalized size = 0. \[ \int \text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.249, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm csch} \left (fx+e\right ) \right ) ^{4} \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname{csch}\left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname{csch}\left (f x + e\right )^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname{csch}\left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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