Optimal. Leaf size=33 \[ i \tanh ^{-1}(\cosh (x))-\frac {i \tanh ^{-1}\left (\frac {\cosh (x)+i \sinh (x)}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {3518, 3110, 3770, 3074, 206} \[ i \tanh ^{-1}(\cosh (x))-\frac {i \tanh ^{-1}\left (\frac {\cosh (x)+i \sinh (x)}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3074
Rule 3110
Rule 3518
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{i+\tanh (x)} \, dx &=\int \frac {\coth (x)}{i \cosh (x)+\sinh (x)} \, dx\\ &=i \int \left (-\text {csch}(x)-\frac {i}{\cosh (x)-i \sinh (x)}\right ) \, dx\\ &=-(i \int \text {csch}(x) \, dx)+\int \frac {1}{\cosh (x)-i \sinh (x)} \, dx\\ &=i \tanh ^{-1}(\cosh (x))+i \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,-\cosh (x)-i \sinh (x)\right )\\ &=i \tanh ^{-1}(\cosh (x))-\frac {i \tanh ^{-1}\left (\frac {\cosh (x)+i \sinh (x)}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 46, normalized size = 1.39 \[ -i \left (\sqrt {2} \tanh ^{-1}\left (\frac {1+i \tanh \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )-\log \left (\cosh \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 43, normalized size = 1.30 \[ -\frac {1}{2} i \, \sqrt {2} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} + e^{x}\right ) + \frac {1}{2} i \, \sqrt {2} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} + e^{x}\right ) + i \, \log \left (e^{x} + 1\right ) - i \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 28, normalized size = 0.85 \[ \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} e^{x}\right ) + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 29, normalized size = 0.88 \[ \sqrt {2}\, \arctan \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2 i\right ) \sqrt {2}}{4}\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 34, normalized size = 1.03 \[ -\sqrt {2} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} e^{\left (-x\right )}\right ) + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 61, normalized size = 1.85 \[ \ln \left (-8\,{\mathrm {e}}^x-8\right )\,1{}\mathrm {i}-\ln \left (8-8\,{\mathrm {e}}^x\right )\,1{}\mathrm {i}-\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^x\,\left (4-4{}\mathrm {i}\right )-\sqrt {2}\,4{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^x\,\left (4-4{}\mathrm {i}\right )+\sqrt {2}\,4{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
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 \[ \int \frac {\operatorname {csch}{\relax (x )}}{\tanh {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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