Optimal. Leaf size=23 \[ 2 i \coth (x)-\tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{\sinh (x)+i} \]
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Rubi [A] time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2768, 2748, 3767, 8, 3770} \[ 2 i \coth (x)-\tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2768
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx &=\frac {\coth (x)}{i+\sinh (x)}+\int \text {csch}^2(x) (-2 i+\sinh (x)) \, dx\\ &=\frac {\coth (x)}{i+\sinh (x)}-2 i \int \text {csch}^2(x) \, dx+\int \text {csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{i+\sinh (x)}-2 \operatorname {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=-\tanh ^{-1}(\cosh (x))+2 i \coth (x)+\frac {\coth (x)}{i+\sinh (x)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 36, normalized size = 1.57 \[ \text {sech}(x) \left (2 i \sinh (x)+i \text {csch}(x)-\sqrt {\cosh ^2(x)} \tanh ^{-1}\left (\sqrt {\cosh ^2(x)}\right )+1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 77, normalized size = 3.35 \[ -\frac {{\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} + 1\right ) - {\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 2 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 4}{e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 44, normalized size = 1.91 \[ \frac {2 \, {\left (e^{\left (2 \, x\right )} + i \, e^{x} - 2\right )}}{e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 1.52 \[ \frac {i \tanh \left (\frac {x}{2}\right )}{2}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )}+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 53, normalized size = 2.30 \[ -\frac {4 \, {\left (-i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 2\right )}}{2 \, e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} - 2 i} - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 51, normalized size = 2.22 \[ \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )+\frac {2\,{\mathrm {e}}^{2\,x}-4+{\mathrm {e}}^x\,2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}+{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\operatorname {csch}^{2}{\relax (x )}}{\sinh {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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