Integrand size = 13, antiderivative size = 47 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {3}{2} \text {arctanh}(\cosh (x))-4 i \coth (x)+\frac {4}{3} i \coth ^3(x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)} \]
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Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2847, 2827, 3852, 3853, 3855} \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {3}{2} \text {arctanh}(\cosh (x))+\frac {4}{3} i \coth ^3(x)-4 i \coth (x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i} \]
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Rule 2827
Rule 2847
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)}+\int \text {csch}^4(x) (-4 i+3 \sinh (x)) \, dx \\ & = \frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)}-4 i \int \text {csch}^4(x) \, dx+3 \int \text {csch}^3(x) \, dx \\ & = -\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)}-\frac {3}{2} \int \text {csch}(x) \, dx+4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right ) \\ & = \frac {3}{2} \text {arctanh}(\cosh (x))-4 i \coth (x)+\frac {4}{3} i \coth ^3(x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.13 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {1}{6} \text {sech}(x) \left (-9+9 \text {arctanh}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)}-8 i \text {csch}(x)-3 \text {csch}^2(x)+2 i \text {csch}^3(x)-16 i \sinh (x)\right ) \]
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Time = 4.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.51
method | result | size |
default | \(-\frac {7 i \tanh \left (\frac {x}{2}\right )}{8}+\frac {i \tanh \left (\frac {x}{2}\right )^{3}}{24}+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {7 i}{8 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}\) | \(71\) |
risch | \(-\frac {9 i {\mathrm e}^{5 x}-24 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{6 x}-24 i {\mathrm e}^{3 x}+39 \,{\mathrm e}^{2 x}+7 i {\mathrm e}^{x}-16}{3 \left ({\mathrm e}^{2 x}-1\right )^{3} \left ({\mathrm e}^{x}+i\right )}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{2}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{2}\) | \(72\) |
parallelrisch | \(\frac {\left (-36 i-36 \tanh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )+i \tanh \left (\frac {x}{2}\right )^{4}-2 i \coth \left (\frac {x}{2}\right )^{2}-18 i \tanh \left (\frac {x}{2}\right )^{2}-\coth \left (\frac {x}{2}\right )^{3}+2 \tanh \left (\frac {x}{2}\right )^{3}-90 i+18 \coth \left (\frac {x}{2}\right )}{24 \tanh \left (\frac {x}{2}\right )+24 i}\) | \(80\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.70 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {9 \, {\left (e^{\left (7 \, x\right )} + i \, e^{\left (6 \, x\right )} - 3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 \, e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} + 1\right ) - 9 \, {\left (e^{\left (7 \, x\right )} + i \, e^{\left (6 \, x\right )} - 3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 \, e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 18 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} + 48 \, e^{\left (4 \, x\right )} + 48 i \, e^{\left (3 \, x\right )} - 78 \, e^{\left (2 \, x\right )} - 14 i \, e^{x} + 32}{6 \, {\left (e^{\left (7 \, x\right )} + i \, e^{\left (6 \, x\right )} - 3 \, e^{\left (5 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 \, e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - e^{x} - i\right )}} \]
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Timed out. \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (35) = 70\).
Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.19 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {-7 i \, e^{\left (-x\right )} + 39 \, e^{\left (-2 \, x\right )} + 24 i \, e^{\left (-3 \, x\right )} - 24 \, e^{\left (-4 \, x\right )} - 9 i \, e^{\left (-5 \, x\right )} + 9 \, e^{\left (-6 \, x\right )} - 16}{3 \, {\left (e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} + i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i\right )}} + \frac {3}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.23 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=-\frac {2}{e^{x} + i} - \frac {3 \, e^{\left (5 \, x\right )} + 6 i \, e^{\left (4 \, x\right )} - 24 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 10 i}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} + \frac {3}{2} \, \log \left (e^{x} + 1\right ) - \frac {3}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 1.49 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.81 \[ \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx=\frac {3\,\ln \left (3\,{\mathrm {e}}^x+3\right )}{2}-\frac {3\,\ln \left (3\,{\mathrm {e}}^x-3\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {4{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {8{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]
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