Optimal. Leaf size=23 \[ -\tanh ^{-1}(\cosh (x))+2 i \coth (x)+\frac {\coth (x)}{i+\sinh (x)} \]
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Rubi [A]
time = 0.04, 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 = {2847, 2827,
3852, 8, 3855} \begin {gather*} 2 i \coth (x)-\tanh ^{-1}(\cosh (x))+\frac {\coth (x)}{\sinh (x)+i} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2827
Rule 2847
Rule 3852
Rule 3855
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 \text {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 = 0.03, size = 36, normalized size = 1.57 \begin {gather*} \text {sech}(x) \left (1-\tanh ^{-1}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)}+i \text {csch}(x)+2 i \sinh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 35, normalized size = 1.52
method | result | size |
default | \(\frac {i \tanh \left (\frac {x}{2}\right )}{2}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )}+\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(35\) |
risch | \(\frac {-4+2 i {\mathrm e}^{x}+2 \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right ) \left ({\mathrm e}^{x}+i\right )}-\ln \left ({\mathrm e}^{x}+1\right )+\ln \left ({\mathrm e}^{x}-1\right )\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
time = 0.27, size = 51, normalized size = 2.22 \begin {gather*} -\frac {2 \, {\left (-i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 2\right )}}{e^{\left (-x\right )} + i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i} - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 77 vs. \(2 (19) = 38\).
time = 0.44, size = 77, normalized size = 3.35 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {csch}^{2}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 44 vs. \(2 (19) = 38\).
time = 0.42, size = 44, normalized size = 1.91 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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