Optimal. Leaf size=125 \[ \frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{p+1}}{b f (2 p+3)}-\frac{(a-b (2 p+3)) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac{b \sinh ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right )}{b f (2 p+3)} \]
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Rubi [A] time = 0.102418, antiderivative size = 119, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3190, 388, 246, 245} \[ \frac{\left (1-\frac{a}{2 b p+3 b}\right ) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac{b \sinh ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right )}{f}+\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{p+1}}{b f (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right )^p \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{\left (1-\frac{a}{3 b+2 b p}\right ) \operatorname{Subst}\left (\int \left (a+b x^2\right )^p \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{\left (\left (1-\frac{a}{3 b+2 b p}\right ) \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 \left (1+\frac{b x^2}{a}\right )^p \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{\left (1-\frac{a}{3 b+2 b p}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right ) \sinh (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}}{f}\\ \end{align*}
Mathematica [A] time = 0.20196, size = 120, normalized size = 0.96 \[ \frac{\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac{b \sinh ^2(e+f x)}{a}+1\right )^{-p} \left ((b (2 p+3)-a) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sinh ^2(e+f x)}{a}\right )+\left (a+b \sinh ^2(e+f x)\right ) \left (\frac{b \sinh ^2(e+f x)}{a}+1\right )^p\right )}{b f (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.453, size = 0, normalized size = 0. \begin{align*} \int \left ( \cosh \left ( fx+e \right ) \right ) ^{3} \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} \cosh \left (f x + e\right )^{3}\,{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} \cosh \left (f x + e\right )^{3}, 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} \cosh \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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