Optimal. Leaf size=74 \[ \frac{a \sinh (x)}{a^2-b^2}-\frac{b \cosh (x)}{a^2-b^2}-\frac{a^2 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]
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Rubi [A] time = 0.0826827, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3099, 3074, 206, 2638} \[ \frac{a \sinh (x)}{a^2-b^2}-\frac{b \cosh (x)}{a^2-b^2}-\frac{a^2 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3099
Rule 3074
Rule 206
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx &=\frac{a \sinh (x)}{a^2-b^2}-\frac{a^2 \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac{b \int \sinh (x) \, dx}{a^2-b^2}\\ &=-\frac{b \cosh (x)}{a^2-b^2}+\frac{a \sinh (x)}{a^2-b^2}-\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{a^2-b^2}\\ &=-\frac{a^2 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac{b \cosh (x)}{a^2-b^2}+\frac{a \sinh (x)}{a^2-b^2}\\ \end{align*}
Mathematica [A] time = 0.193209, size = 89, normalized size = 1.2 \[ \frac{a \left (\sqrt{a-b} (a+b) \sinh (x)-2 a \sqrt{a+b} \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )\right )-b \sqrt{a-b} (a+b) \cosh (x)}{(a-b)^{3/2} (a+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 93, normalized size = 1.3 \begin{align*} -2\,{\frac{{a}^{2}}{ \left ( a+b \right ) \left ( a-b \right ) \sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-8\,{\frac{1}{ \left ( 8\,a+8\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) }}-8\,{\frac{1}{ \left ( 8\,a-8\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88034, size = 1106, normalized size = 14.95 \begin{align*} \left [-\frac{a^{3} + a^{2} b - a b^{2} - b^{3} -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} - 2 \,{\left (a^{2} \cosh \left (x\right ) + a^{2} \sinh \left (x\right )\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{-a^{2} + b^{2}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{2 \,{\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) +{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}, -\frac{a^{3} + a^{2} b - a b^{2} - b^{3} -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} - 4 \,{\left (a^{2} \cosh \left (x\right ) + a^{2} \sinh \left (x\right )\right )} \sqrt{a^{2} - b^{2}} \arctan \left (\frac{\sqrt{a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )}\right )}{2 \,{\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) +{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}\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 = 1.14338, size = 82, normalized size = 1.11 \begin{align*} -\frac{2 \, a^{2} \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac{3}{2}}} - \frac{e^{\left (-x\right )}}{2 \,{\left (a - b\right )}} + \frac{e^{x}}{2 \,{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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