Optimal. Leaf size=101 \[ \frac{a^2 b x}{\left (a^2-b^2\right )^2}+\frac{b x}{2 \left (a^2-b^2\right )}+\frac{a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac{b \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )}-\frac{a^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2} \]
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Rubi [A] time = 0.132672, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3099, 3097, 3133, 2635, 8} \[ \frac{a^2 b x}{\left (a^2-b^2\right )^2}+\frac{b x}{2 \left (a^2-b^2\right )}+\frac{a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac{b \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )}-\frac{a^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3099
Rule 3097
Rule 3133
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx &=\frac{a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac{a^2 \int \frac{\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac{b \int \sinh ^2(x) \, dx}{a^2-b^2}\\ &=\frac{a^2 b x}{\left (a^2-b^2\right )^2}-\frac{b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )}+\frac{a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac{\left (i a^3\right ) \int \frac{-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac{b \int 1 \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac{a^2 b x}{\left (a^2-b^2\right )^2}+\frac{b x}{2 \left (a^2-b^2\right )}-\frac{a^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}-\frac{b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )}+\frac{a \sinh ^2(x)}{2 \left (a^2-b^2\right )}\\ \end{align*}
Mathematica [A] time = 0.144926, size = 75, normalized size = 0.74 \[ \frac{\left (b^3-a^2 b\right ) \sinh (2 x)+a \left (a^2-b^2\right ) \cosh (2 x)+6 a^2 b x-4 a^3 \log (a \cosh (x)+b \sinh (x))-2 b^3 x}{4 (a-b)^2 (a+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 175, normalized size = 1.7 \begin{align*} -16\,{\frac{1}{ \left ( 32\,a-32\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) }}+8\,{\frac{1}{ \left ( 16\,a-16\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{2}}}+{\frac{a}{ \left ( a-b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{b}{2\, \left ( a-b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{{a}^{3}}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }+8\,{\frac{1}{ \left ( 16\,a+16\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{2}}}+16\,{\frac{1}{ \left ( 32\,a+32\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) }}+{\frac{a}{ \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{b}{2\, \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14197, size = 117, normalized size = 1.16 \begin{align*} -\frac{a^{3} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (2 \, a + b\right )} x}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac{e^{\left (2 \, x\right )}}{8 \,{\left (a + b\right )}} + \frac{e^{\left (-2 \, x\right )}}{8 \,{\left (a - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88408, size = 818, normalized size = 8.1 \begin{align*} \frac{{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{4} + 4 \,{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{4} + 4 \,{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} x \cosh \left (x\right )^{2} + a^{3} + a^{2} b - a b^{2} - b^{3} + 2 \,{\left (3 \,{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} x\right )} \sinh \left (x\right )^{2} - 8 \,{\left (a^{3} \cosh \left (x\right )^{2} + 2 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right ) + a^{3} \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \,{\left ({\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{3} + 2 \,{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )}{8 \,{\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \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 = 1.15356, size = 154, normalized size = 1.52 \begin{align*} -\frac{a^{3} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (2 \, a - b\right )} x}{2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac{{\left (4 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-2 \, x\right )}}{8 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac{e^{\left (2 \, x\right )}}{8 \,{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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