Optimal. Leaf size=7 \[ \frac{1}{2} \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.013489, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4288, 3770} \[ \frac{1}{2} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 4288
Rule 3770
Rubi steps
\begin{align*} \int \text{csch}(2 x) \sinh (x) \, dx &=\frac{1}{2} \int \text{sech}(x) \, dx\\ &=\frac{1}{2} \tan ^{-1}(\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0028293, size = 7, normalized size = 1. \[ \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 4, normalized size = 0.6 \begin{align*} \arctan \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57405, size = 9, normalized size = 1.29 \begin{align*} -\arctan \left (e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07055, size = 36, normalized size = 5.14 \begin{align*} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{csch}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20529, size = 4, normalized size = 0.57 \begin{align*} \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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