Optimal. Leaf size=12 \[ -\tanh ^{-1}(\cosh (x))+i \coth (x) \]
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Rubi [A] time = 0.0444647, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 3767, 8, 3770} \[ -\tanh ^{-1}(\cosh (x))+i \coth (x) \]
Antiderivative was successfully verified.
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Rule 2706
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{i+\sinh (x)} \, dx &=-\left (i \int \text{csch}^2(x) \, dx\right )+\int \text{csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))-\operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=-\tanh ^{-1}(\cosh (x))+i \coth (x)\\ \end{align*}
Mathematica [B] time = 0.0346959, size = 32, normalized size = 2.67 \[ \frac{1}{2} i \tanh \left (\frac{x}{2}\right )+\frac{1}{2} i \coth \left (\frac{x}{2}\right )+\log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 23, normalized size = 1.9 \begin{align*}{\frac{i}{2}}\tanh \left ({\frac{x}{2}} \right ) +{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07307, size = 36, normalized size = 3. \begin{align*} -\frac{2 i}{e^{\left (-2 \, x\right )} - 1} - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0131, size = 108, normalized size = 9. \begin{align*} -\frac{{\left (e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} + 1\right ) -{\left (e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} - 1\right ) - 2 i}{e^{\left (2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.217057, size = 22, normalized size = 1.83 \begin{align*} \log{\left (e^{x} - 1 \right )} - \log{\left (e^{x} + 1 \right )} + \frac{2 i}{e^{2 x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12792, size = 32, normalized size = 2.67 \begin{align*} \frac{2 i}{e^{\left (2 \, x\right )} - 1} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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