Optimal. Leaf size=15 \[ -\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x) \]
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Rubi [A] time = 0.06, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2706, 2606, 30, 8} \[ -\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2606
Rule 2706
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx &=-\left (i \int \coth (x) \text {csch}^2(x) \, dx\right )+\int \coth (x) \text {csch}(x) \, dx\\ &=-(i \operatorname {Subst}(\int 1 \, dx,x,-i \text {csch}(x)))-i \operatorname {Subst}(\int x \, dx,x,-i \text {csch}(x))\\ &=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ -\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 33, normalized size = 2.20 \[ -\frac {2 \, e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} - 2 \, e^{x}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 24, normalized size = 1.60 \[ \frac {2 \, e^{\left (-x\right )} - 2 \, e^{x} + 2 i}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 34, normalized size = 2.27 \[ \frac {\tanh \left (\frac {x}{2}\right )}{2}+\frac {i \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 67, normalized size = 4.47 \[ \frac {2 \, e^{\left (-x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 25, normalized size = 1.67 \[ \frac {2\,{\mathrm {e}}^x\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 32, normalized size = 2.13 \[ \frac {- 2 e^{3 x} + 2 i e^{2 x} + 2 e^{x}}{e^{4 x} - 2 e^{2 x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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