Optimal. Leaf size=26 \[ -\coth (x)+i \tanh ^{-1}(\cosh (x))-\frac {i \coth (x)}{\text {csch}(x)+i} \]
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Rubi [A] time = 0.08, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 = {3790, 3789, 3770, 3794} \[ -\coth (x)+i \tanh ^{-1}(\cosh (x))-\frac {i \coth (x)}{\text {csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3789
Rule 3790
Rule 3794
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{i+\text {csch}(x)} \, dx &=-\coth (x)-i \int \frac {\text {csch}^2(x)}{i+\text {csch}(x)} \, dx\\ &=-\coth (x)-i \int \text {csch}(x) \, dx-\int \frac {\text {csch}(x)}{i+\text {csch}(x)} \, dx\\ &=i \tanh ^{-1}(\cosh (x))-\coth (x)-\frac {i \coth (x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 70, normalized size = 2.69 \[ -\frac {1}{2} \tanh \left (\frac {x}{2}\right )-\frac {1}{2} \coth \left (\frac {x}{2}\right )-i \log \left (\sinh \left (\frac {x}{2}\right )\right )+i \log \left (\cosh \left (\frac {x}{2}\right )\right )-\frac {2 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 77, normalized size = 2.96 \[ \frac {{\left (i \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + {\left (-i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 4 i}{e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )} - e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 46, normalized size = 1.77 \[ \frac {2 \, {\left (e^{\left (2 \, x\right )} - i \, e^{x} - 2\right )}}{i \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - i \, e^{x} - 1} + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 35, normalized size = 1.35 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {2}{\tanh \left (\frac {x}{2}\right )-i}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-i \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 55, normalized size = 2.12 \[ -\frac {8 \, {\left (e^{\left (-x\right )} - i \, e^{\left (-2 \, x\right )} + 2 i\right )}}{4 \, e^{\left (-x\right )} - 4 i \, e^{\left (-2 \, x\right )} - 4 \, e^{\left (-3 \, x\right )} + 4 i} + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 60, normalized size = 2.31 \[ -\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}-2{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}+2{}\mathrm {i}\right )\,1{}\mathrm {i}+\frac {{\mathrm {e}}^{2\,x}\,2{}\mathrm {i}+2\,{\mathrm {e}}^x-4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}-{\mathrm {e}}^{3\,x}+{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\operatorname {csch}^{3}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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