Optimal. Leaf size=10 \[ \sinh (x)-\frac{1}{2} \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.0230417, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {388, 203} \[ \sinh (x)-\frac{1}{2} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 388
Rule 203
Rubi steps
\begin{align*} \int \coth (2 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1+2 x^2}{2+2 x^2} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\operatorname{Subst}\left (\int \frac{1}{2+2 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{2} \tan ^{-1}(\sinh (x))+\sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0084669, size = 10, normalized size = 1. \[ \sinh (x)-\frac{1}{2} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 9, normalized size = 0.9 \begin{align*} \sinh \left ( x \right ) -\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 = 1.51668, size = 22, normalized size = 2.2 \begin{align*} \arctan \left (e^{\left (-x\right )}\right ) - \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05331, size = 167, normalized size = 16.7 \begin{align*} -\frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + 1}{2 \,{\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 )} \coth{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15817, size = 22, normalized size = 2.2 \begin{align*} -\arctan \left (e^{x}\right ) - \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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