Optimal. Leaf size=11 \[ -\tanh ^{-1}(\cosh (x))-i x \]
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Rubi [A] time = 0.0383253, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3888, 3770} \[ -\tanh ^{-1}(\cosh (x))-i x \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3770
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{i+\text{csch}(x)} \, dx &=\int (-i+\text{csch}(x)) \, dx\\ &=-i x+\int \text{csch}(x) \, dx\\ &=-i x-\tanh ^{-1}(\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0329229, size = 13, normalized size = 1.18 \[ \log \left (\tanh \left (\frac{x}{2}\right )\right )-i x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 27, normalized size = 2.5 \begin{align*} -i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) +i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00286, size = 27, normalized size = 2.45 \begin{align*} -i \, x - \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 [A] time = 1.61851, size = 49, normalized size = 4.45 \begin{align*} -i \, x - \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\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 \frac{\coth ^{2}{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19109, size = 23, normalized size = 2.09 \begin{align*} -i \, x - \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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