Optimal. Leaf size=128 \[ \frac{d \cosh (e+f x) \left (-\sinh ^2(e+f x)\right )^{\frac{1-m}{2}} (d \sinh (e+f x))^{m-1} \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac{b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right )}{f} \]
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Rubi [A] time = 0.116103, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3189, 430, 429} \[ \frac{d \cosh (e+f x) \left (-\sinh ^2(e+f x)\right )^{\frac{1-m}{2}} (d \sinh (e+f x))^{m-1} \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac{b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};\cosh ^2(e+f x),-\frac{b \cosh ^2(e+f x)}{a-b}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3189
Rule 430
Rule 429
Rubi steps
\begin{align*} \int (d \sinh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx &=\frac{\left (d (d \sinh (e+f x))^{2 \left (-\frac{1}{2}+\frac{m}{2}\right )} \left (-\sinh ^2(e+f x)\right )^{\frac{1}{2}-\frac{m}{2}}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right )^{\frac{1}{2} (-1+m)} \left (a-b+b x^2\right )^p \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac{\left (d \left (a-b+b \cosh ^2(e+f x)\right )^p \left (1+\frac{b \cosh ^2(e+f x)}{a-b}\right )^{-p} (d \sinh (e+f x))^{2 \left (-\frac{1}{2}+\frac{m}{2}\right )} \left (-\sinh ^2(e+f x)\right )^{\frac{1}{2}-\frac{m}{2}}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right )^{\frac{1}{2} (-1+m)} \left (1+\frac{b x^2}{a-b}\right )^p \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac{d F_1\left (\frac{1}{2};\frac{1-m}{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} (d \sinh (e+f x))^{-1+m} \left (-\sinh ^2(e+f x)\right )^{\frac{1-m}{2}}}{f}\\ \end{align*}
Mathematica [F] time = 8.65666, size = 0, normalized size = 0. \[ \int (d \sinh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.641, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sinh \left ( fx+e \right ) \right ) ^{m} \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} \left (d \sinh \left (f x + e\right )\right )^{m}\,{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} \left (d \sinh \left (f x + e\right )\right )^{m}, 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} \left (d \sinh \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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