Optimal. Leaf size=12 \[ -\text {csch}(x)-i \log (\sinh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3879, 43} \[ -\text {csch}(x)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx &=\operatorname {Subst}\left (\int \frac {i-i x}{x^2} \, dx,x,i \sinh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {i}{x^2}-\frac {i}{x}\right ) \, dx,x,i \sinh (x)\right )\\ &=-\text {csch}(x)-i \log (\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ -\text {csch}(x)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 40, normalized size = 3.33 \[ \frac {i \, x e^{\left (2 \, x\right )} + {\left (-i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - i \, x - 2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 38, normalized size = 3.17 \[ \frac {i \, e^{\left (-x\right )} - i \, e^{x} + 2}{e^{\left (-x\right )} - e^{x}} - i \, \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 12, normalized size = 1.00 \[ -\mathrm {csch}\relax (x )+i \ln \left (\mathrm {csch}\relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 36, normalized size = 3.00 \[ -i \, x + \frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 27, normalized size = 2.25 \[ -\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}+x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,1{}\mathrm {i} \]
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 \[ \int \frac {\coth ^{3}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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