Optimal. Leaf size=83 \[ \frac{\left (a^2-2 b^2\right ) \cosh ^2(x)}{2 b^3}-\frac{a \left (a^2-2 b^2\right ) \cosh (x)}{b^4}+\frac{\left (a^2-b^2\right )^2 \log (a+b \cosh (x))}{b^5}-\frac{a \cosh ^3(x)}{3 b^2}+\frac{\cosh ^4(x)}{4 b} \]
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Rubi [A] time = 0.107238, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2668, 697} \[ \frac{\left (a^2-2 b^2\right ) \cosh ^2(x)}{2 b^3}-\frac{a \left (a^2-2 b^2\right ) \cosh (x)}{b^4}+\frac{\left (a^2-b^2\right )^2 \log (a+b \cosh (x))}{b^5}-\frac{a \cosh ^3(x)}{3 b^2}+\frac{\cosh ^4(x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\sinh ^5(x)}{a+b \cosh (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{a+x} \, dx,x,b \cosh (x)\right )}{b^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^3 \left (1-\frac{2 b^2}{a^2}\right )+\left (a^2-2 b^2\right ) x-a x^2+x^3+\frac{\left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \cosh (x)\right )}{b^5}\\ &=-\frac{a \left (a^2-2 b^2\right ) \cosh (x)}{b^4}+\frac{\left (a^2-2 b^2\right ) \cosh ^2(x)}{2 b^3}-\frac{a \cosh ^3(x)}{3 b^2}+\frac{\cosh ^4(x)}{4 b}+\frac{\left (a^2-b^2\right )^2 \log (a+b \cosh (x))}{b^5}\\ \end{align*}
Mathematica [A] time = 0.107378, size = 84, normalized size = 1.01 \[ \frac{-12 b^2 \left (3 b^2-2 a^2\right ) \cosh (2 x)-24 a b \left (4 a^2-7 b^2\right ) \cosh (x)+96 \left (a^2-b^2\right )^2 \log (a+b \cosh (x))-8 a b^3 \cosh (3 x)+3 b^4 \cosh (4 x)}{96 b^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 599, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0499, size = 240, normalized size = 2.89 \begin{align*} -\frac{{\left (8 \, a b^{2} e^{\left (-x\right )} - 3 \, b^{3} - 12 \,{\left (2 \, a^{2} b - 3 \, b^{3}\right )} e^{\left (-2 \, x\right )} + 24 \,{\left (4 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-3 \, x\right )}\right )} e^{\left (4 \, x\right )}}{192 \, b^{4}} - \frac{8 \, a b^{2} e^{\left (-3 \, x\right )} - 3 \, b^{3} e^{\left (-4 \, x\right )} + 24 \,{\left (4 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-x\right )} - 12 \,{\left (2 \, a^{2} b - 3 \, b^{3}\right )} e^{\left (-2 \, x\right )}}{192 \, b^{4}} + \frac{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x}{b^{5}} + \frac{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.93527, size = 2228, normalized size = 26.84 \begin{align*} \text{result too large to display} \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 [A] time = 1.22426, size = 167, normalized size = 2.01 \begin{align*} \frac{3 \, b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 8 \, a b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 24 \, a^{2} b{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 48 \, b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 96 \, a^{3}{\left (e^{\left (-x\right )} + e^{x}\right )} + 192 \, a b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}}{192 \, b^{4}} + \frac{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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