Optimal. Leaf size=38 \[ \frac{\left (a^2+b^2\right ) \log (a+b \sinh (x))}{b^3}-\frac{a \sinh (x)}{b^2}+\frac{\sinh ^2(x)}{2 b} \]
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Rubi [A] time = 0.0611395, antiderivative size = 38, 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} \[ \frac{\left (a^2+b^2\right ) \log (a+b \sinh (x))}{b^3}-\frac{a \sinh (x)}{b^2}+\frac{\sinh ^2(x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{a+b \sinh (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{-b^2-x^2}{a+x} \, dx,x,b \sinh (x)\right )}{b^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a-x+\frac{-a^2-b^2}{a+x}\right ) \, dx,x,b \sinh (x)\right )}{b^3}\\ &=\frac{\left (a^2+b^2\right ) \log (a+b \sinh (x))}{b^3}-\frac{a \sinh (x)}{b^2}+\frac{\sinh ^2(x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0348453, size = 38, normalized size = 1. \[ -\frac{-\left (a^2+b^2\right ) \log (a+b \sinh (x))+a b \sinh (x)-\frac{1}{2} b^2 \sinh ^2(x)}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 185, normalized size = 4.9 \begin{align*}{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{a}{{b}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{{b}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{{a}^{2}}{{b}^{3}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }+{\frac{1}{b}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.29438, size = 109, normalized size = 2.87 \begin{align*} -\frac{{\left (4 \, a e^{\left (-x\right )} - b\right )} e^{\left (2 \, x\right )}}{8 \, b^{2}} + \frac{4 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )}}{8 \, b^{2}} + \frac{{\left (a^{2} + b^{2}\right )} x}{b^{3}} + \frac{{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80766, size = 629, normalized size = 16.55 \begin{align*} \frac{b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} - 4 \, a b \cosh \left (x\right )^{3} - 8 \,{\left (a^{2} + b^{2}\right )} x \cosh \left (x\right )^{2} + 4 \,{\left (b^{2} \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )^{3} + 4 \, a b \cosh \left (x\right ) + 2 \,{\left (3 \, b^{2} \cosh \left (x\right )^{2} - 6 \, a b \cosh \left (x\right ) - 4 \,{\left (a^{2} + b^{2}\right )} x\right )} \sinh \left (x\right )^{2} + b^{2} + 8 \,{\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \,{\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \,{\left (b^{2} \cosh \left (x\right )^{3} - 3 \, a b \cosh \left (x\right )^{2} - 4 \,{\left (a^{2} + b^{2}\right )} x \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{8 \,{\left (b^{3} \cosh \left (x\right )^{2} + 2 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right ) + b^{3} \sinh \left (x\right )^{2}\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 [A] time = 1.15903, size = 82, normalized size = 2.16 \begin{align*} \frac{b{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, a{\left (e^{\left (-x\right )} - e^{x}\right )}}{8 \, b^{2}} + \frac{{\left (a^{2} + b^{2}\right )} \log \left ({\left | -b{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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