Optimal. Leaf size=66 \[ -\frac{a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}-\frac{b \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]
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Rubi [A] time = 0.0626764, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3154, 3074, 206} \[ -\frac{a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}-\frac{b \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3154
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=-\frac{a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}-\frac{b \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}\\ &=-\frac{a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}-\frac{(i b) \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{b \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{a}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.189291, size = 125, normalized size = 1.89 \[ -\frac{2 b^2 \sqrt{a+b} \sinh (x) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )+2 a b \sqrt{a+b} \cosh (x) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )+a \sqrt{a-b} (a+b)}{(a-b)^{3/2} (a+b)^2 (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 99, normalized size = 1.5 \begin{align*} 4\,{\frac{-2\,\tanh \left ( x/2 \right ) b-2\,a}{ \left ( 4\,{a}^{2}-4\,{b}^{2} \right ) \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}-8\,{\frac{b}{ \left ( 4\,{a}^{2}-4\,{b}^{2} \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 ) } \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 = 1.9003, size = 1473, normalized size = 22.32 \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 = 1.1297, size = 97, normalized size = 1.47 \begin{align*} -\frac{2 \, b \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{2 \, a e^{x}}{{\left (a^{2} - b^{2}\right )}{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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