Optimal. Leaf size=17 \[ -\tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{\text {csch}(x)+i} \]
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Rubi [A] time = 0.05, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3789, 3770, 3794} \[ -\tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{\text {csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3789
Rule 3794
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{i+\text {csch}(x)} \, dx &=-\left (i \int \frac {\text {csch}(x)}{i+\text {csch}(x)} \, dx\right )+\int \text {csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 37, normalized size = 2.18 \[ \log \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {2 i \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 31, normalized size = 1.82 \[ -\frac {{\left (e^{x} - i\right )} \log \left (e^{x} + 1\right ) - {\left (e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 2}{e^{x} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 22, normalized size = 1.29 \[ \frac {2}{e^{x} - i} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 19, normalized size = 1.12 \[ -\frac {2 i}{\tanh \left (\frac {x}{2}\right )-i}+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 1.71 \[ \frac {4}{2 \, e^{\left (-x\right )} + 2 i} - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 26, normalized size = 1.53 \[ \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )+\frac {2}{{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\operatorname {csch}^{2}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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