Optimal. Leaf size=13 \[ -i \log (-\sinh (x)+i) \]
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Rubi [A] time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3879, 31} \[ -i \log (-\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 31
Rule 3879
Rubi steps
\begin {align*} \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{i+i x} \, dx,x,i \sinh (x)\right )\\ &=-i \log (i-\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ -i \log (-\sinh (x)+i) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 11, normalized size = 0.85 \[ i \, x - 2 i \, \log \left (e^{x} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 13, normalized size = 1.00 \[ i \, x - 2 i \, \log \left (i \, e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 17, normalized size = 1.31 \[ -i \ln \left (i+\mathrm {csch}\relax (x )\right )+i \ln \left (\mathrm {csch}\relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 15, normalized size = 1.15 \[ -i \, x - 2 i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 14, normalized size = 1.08 \[ x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,2{}\mathrm {i} \]
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 \[ \int \frac {\coth {\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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