Integrand size = 13, antiderivative size = 29 \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=2 i \text {csch}(x)+\frac {\text {csch}^2(x)}{2}+2 \log (\sinh (x))-2 \log (i+\sinh (x)) \]
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Time = 0.03 (sec) , antiderivative size = 29, 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 = {2786, 78} \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {\text {csch}^2(x)}{2}+2 i \text {csch}(x)+2 \log (\sinh (x))-2 \log (\sinh (x)+i) \]
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Rule 78
Rule 2786
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {i-x}{x^3 (i+x)} \, dx,x,\sinh (x)\right ) \\ & = -\text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {2 i}{x^2}-\frac {2}{x}+\frac {2}{i+x}\right ) \, dx,x,\sinh (x)\right ) \\ & = 2 i \text {csch}(x)+\frac {\text {csch}^2(x)}{2}+2 \log (\sinh (x))-2 \log (i+\sinh (x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=2 i \text {csch}(x)+\frac {\text {csch}^2(x)}{2}+2 \log (\sinh (x))-2 \log (i+\sinh (x)) \]
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Time = 21.47 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55
method | result | size |
risch | \(\frac {2 i {\mathrm e}^{x} \left (2 \,{\mathrm e}^{2 x}-2-i {\mathrm e}^{x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+2 \ln \left ({\mathrm e}^{2 x}-1\right )-4 \ln \left ({\mathrm e}^{x}+i\right )\) | \(45\) |
default | \(-i \tanh \left (\frac {x}{2}\right )+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}-4 \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )+\frac {i}{\tanh \left (\frac {x}{2}\right )}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+2 \ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(51\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41 \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {2 \, {\left ({\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 2 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} + i\right ) + 2 i \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - 2 i \, e^{x}\right )}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \]
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Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {4 i e^{3 x} + 2 e^{2 x} - 4 i e^{x}}{e^{4 x} - 2 e^{2 x} + 1} - 4 \log {\left (e^{x} + i \right )} + 2 \log {\left (e^{2 x} - 1 \right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (23) = 46\).
Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (2 i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 2 i \, e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) - 4 \, \log \left (e^{\left (-x\right )} - i\right ) + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (-2 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + 2 i \, e^{x}\right )}}{{\left (e^{x} + 1\right )}^{2} {\left (e^{x} - 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) - 4 \, \log \left (e^{x} + i\right ) + 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {\coth ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+2\,\ln \left (-{\mathrm {e}}^{2\,x}\,6{}\mathrm {i}+6{}\mathrm {i}\right )-4\,\ln \left (144\,{\mathrm {e}}^x+144{}\mathrm {i}\right )+\frac {2+{\mathrm {e}}^x\,4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]
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