Integrand size = 13, antiderivative size = 15 \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x) \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 2686, 30, 8} \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x) \]
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Rule 8
Rule 30
Rule 2686
Rule 2785
Rubi steps \begin{align*} \text {integral}& = -\left (i \int \coth (x) \text {csch}^2(x) \, dx\right )+\int \coth (x) \text {csch}(x) \, dx \\ & = -(i \text {Subst}(\int 1 \, dx,x,-i \text {csch}(x)))-i \text {Subst}(\int x \, dx,x,-i \text {csch}(x)) \\ & = -\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x) \]
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Time = 9.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{x} \left (-i {\mathrm e}^{x}+{\mathrm e}^{2 x}-1\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}\) | \(24\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{2}+\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}}\) | \(34\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (11) = 22\).
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=-\frac {2 \, {\left (e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )} - e^{x}\right )}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (10) = 20\).
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.13 \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=\frac {- 2 e^{3 x} + 2 i e^{2 x} + 2 e^{x}}{e^{4 x} - 2 e^{2 x} + 1} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (11) = 22\).
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 4.47 \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=\frac {2 \, e^{\left (-x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.53 \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=\frac {2 \, {\left (e^{\left (-x\right )} - e^{x} + i\right )}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \]
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Time = 1.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \frac {\coth ^3(x)}{i+\sinh (x)} \, dx=\frac {2\,{\mathrm {e}}^x\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2} \]
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