Integrand size = 13, antiderivative size = 37 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3}{2} i \text {arctanh}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2847, 2827, 3853, 3855, 3852, 8} \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3}{2} i \text {arctanh}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{\sinh (x)+i} \]
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Rule 8
Rule 2827
Rule 2847
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}+\int \text {csch}^3(x) (-3 i+2 \sinh (x)) \, dx \\ & = \frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}-3 i \int \text {csch}^3(x) \, dx+2 \int \text {csch}^2(x) \, dx \\ & = \frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}+\frac {3}{2} i \int \text {csch}(x) \, dx-2 i \text {Subst}(\int 1 \, dx,x,-i \coth (x)) \\ & = -\frac {3}{2} i \text {arctanh}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\frac {1}{2} i \left (4 i+3 \text {csch}(x)-3 \text {arctanh}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)} \text {csch}(x)+2 i \text {csch}^2(x)+\text {csch}^3(x)\right ) \tanh (x) \]
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Time = 3.50 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {2}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}\) | \(53\) |
risch | \(\frac {i \left (3 \,{\mathrm e}^{4 x}-5 \,{\mathrm e}^{2 x}+3 i {\mathrm e}^{3 x}+4-i {\mathrm e}^{x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left ({\mathrm e}^{x}+i\right )}+\frac {3 i \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {3 i \ln \left ({\mathrm e}^{x}+1\right )}{2}\) | \(62\) |
parallelrisch | \(\frac {\left (12 i \tanh \left (\frac {x}{2}\right )-12\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )-i \tanh \left (\frac {x}{2}\right )^{3}-3 i \coth \left (\frac {x}{2}\right )-\coth \left (\frac {x}{2}\right )^{2}-3 \tanh \left (\frac {x}{2}\right )^{2}-24}{8 \tanh \left (\frac {x}{2}\right )+8 i}\) | \(62\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.41 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3 \, {\left (i \, e^{\left (5 \, x\right )} - e^{\left (4 \, x\right )} - 2 i \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + 3 \, {\left (-i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} + 2 i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} - i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 6 i \, e^{\left (4 \, x\right )} + 6 \, e^{\left (3 \, x\right )} + 10 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} - 8 i}{2 \, {\left (e^{\left (5 \, x\right )} + i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + i\right )}} \]
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\[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (27) = 54\).
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.03 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {e^{\left (-x\right )} + 5 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i \, e^{\left (-4 \, x\right )} - 4 i}{e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} - i \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} - i} - \frac {3}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\frac {i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + \frac {2 i}{e^{x} + i} - \frac {3}{2} i \, \log \left (e^{x} + 1\right ) + \frac {3}{2} i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 1.45 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.89 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}-3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}+3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {{\mathrm {e}}^x\,2{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {-2+{\mathrm {e}}^x\,1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]
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