Optimal. Leaf size=37 \[ 2 i \coth (x)+\frac {3}{2} \tanh ^{-1}(\cosh (x))+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}-\frac {3}{2} \coth (x) \text {csch}(x) \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3818, 3787, 3767, 8, 3768, 3770} \[ 2 i \coth (x)+\frac {3}{2} \tanh ^{-1}(\cosh (x))+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}-\frac {3}{2} \coth (x) \text {csch}(x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3768
Rule 3770
Rule 3787
Rule 3818
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx &=\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)}-\int (2 i-3 \text {csch}(x)) \text {csch}^2(x) \, dx\\ &=\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)}-2 i \int \text {csch}^2(x) \, dx+3 \int \text {csch}^3(x) \, dx\\ &=-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)}-\frac {3}{2} \int \text {csch}(x) \, dx-2 \operatorname {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=\frac {3}{2} \tanh ^{-1}(\cosh (x))+2 i \coth (x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [B] time = 0.33, size = 81, normalized size = 2.19 \[ \frac {1}{8} \left (4 i \tanh \left (\frac {x}{2}\right )+4 i \coth \left (\frac {x}{2}\right )-\text {csch}^2\left (\frac {x}{2}\right )-\text {sech}^2\left (\frac {x}{2}\right )-12 \log \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {16 \sinh \left (\frac {x}{2}\right )}{\sinh \left (\frac {x}{2}\right )-i \cosh \left (\frac {x}{2}\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 130, normalized size = 3.51 \[ \frac {{\left (3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} - 6 \, e^{\left (3 \, x\right )} + 6 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 3 i\right )} \log \left (e^{x} + 1\right ) - {\left (3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} - 6 \, e^{\left (3 \, x\right )} + 6 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 3 i\right )} \log \left (e^{x} - 1\right ) - 6 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} - 8}{2 \, e^{\left (5 \, x\right )} - 2 i \, e^{\left (4 \, x\right )} - 4 \, e^{\left (3 \, x\right )} + 4 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - 2 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 50, normalized size = 1.35 \[ -\frac {e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + 2 i}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} - \frac {2 i}{i \, e^{x} + 1} + \frac {3}{2} \, \log \left (e^{x} + 1\right ) - \frac {3}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 53, normalized size = 1.43 \[ \frac {i \tanh \left (\frac {x}{2}\right )}{2}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )-i}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 81, normalized size = 2.19 \[ -\frac {16 \, {\left (-i \, e^{\left (-x\right )} - 5 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 4\right )}}{16 \, e^{\left (-x\right )} - 32 i \, e^{\left (-2 \, x\right )} - 32 \, e^{\left (-3 \, x\right )} + 16 i \, e^{\left (-4 \, x\right )} + 16 \, e^{\left (-5 \, x\right )} + 16 i} + \frac {3}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 63, normalized size = 1.70 \[ \frac {3\,\ln \left (3\,{\mathrm {e}}^x+3\right )}{2}-\frac {3\,\ln \left (3\,{\mathrm {e}}^x-3\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {2}{{\mathrm {e}}^x-\mathrm {i}}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]
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 \[ \int \frac {\operatorname {csch}^{4}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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