Optimal. Leaf size=7 \[ \frac {1}{2} \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 7, normalized size of antiderivative = 1.00, 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 3770
Rule 4288
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*}
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Mathematica [A] time = 0.00, size = 7, normalized size = 1.00 \[ \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 6, normalized size = 0.86 \[ \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 3, normalized size = 0.43 \[ \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 4, normalized size = 0.57 \[ \arctan \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 7, normalized size = 1.00 \[ -\arctan \left (e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 3, normalized size = 0.43 \[ \mathrm {atan}\left ({\mathrm {e}}^x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {csch}{\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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