Integrand size = 13, antiderivative size = 12 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)-i \log (\sinh (x)) \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3964, 45} \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)-i \log (\sinh (x)) \]
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Rule 45
Rule 3964
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {i-i x}{x^2} \, dx,x,i \sinh (x)\right ) \\ & = \text {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*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)-i \log (\sinh (x)) \]
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Time = 2.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\operatorname {csch}\left (x \right )+i \ln \left (\operatorname {csch}\left (x \right )\right )\) | \(12\) |
default | \(-\operatorname {csch}\left (x \right )+i \ln \left (\operatorname {csch}\left (x \right )\right )\) | \(12\) |
risch | \(i x -\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}-1}-i \ln \left ({\mathrm e}^{2 x}-1\right )\) | \(28\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (10) = 20\).
Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.33 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=\frac {i \, x e^{\left (2 \, x\right )} + {\left (-i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - i \, x - 2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} \]
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\[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (10) = 20\).
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 3.00 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-i \, x + \frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (10) = 20\).
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 3.25 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {-i \, e^{\left (-x\right )} + i \, e^{x} - 2}{e^{\left (-x\right )} - e^{x}} - i \, \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right ) \]
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Time = 2.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \frac {\coth ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}+x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,1{}\mathrm {i} \]
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