Integrand size = 13, antiderivative size = 36 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=-\frac {3}{8} \text {arctanh}(\cosh (x))+\frac {1}{5} i \coth ^5(x)-\frac {3}{8} \coth (x) \text {csch}(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x) \]
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Time = 0.07 (sec) , antiderivative size = 36, 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 = {2785, 2687, 30, 2691, 3855} \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=-\frac {3}{8} \text {arctanh}(\cosh (x))+\frac {1}{5} i \coth ^5(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x)-\frac {3}{8} \coth (x) \text {csch}(x) \]
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Rule 30
Rule 2687
Rule 2691
Rule 2785
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\left (i \int \coth ^4(x) \text {csch}^2(x) \, dx\right )+\int \coth ^4(x) \text {csch}(x) \, dx \\ & = -\frac {1}{4} \coth ^3(x) \text {csch}(x)+\frac {3}{4} \int \coth ^2(x) \text {csch}(x) \, dx+\text {Subst}\left (\int x^4 \, dx,x,i \coth (x)\right ) \\ & = \frac {1}{5} i \coth ^5(x)-\frac {3}{8} \coth (x) \text {csch}(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x)+\frac {3}{8} \int \text {csch}(x) \, dx \\ & = -\frac {3}{8} \text {arctanh}(\cosh (x))+\frac {1}{5} i \coth ^5(x)-\frac {3}{8} \coth (x) \text {csch}(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(36)=72\).
Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 4.86 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {1}{10} i \coth \left (\frac {x}{2}\right )-\frac {5}{32} \text {csch}^2\left (\frac {x}{2}\right )+\frac {7}{160} i \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{64} \text {csch}^4\left (\frac {x}{2}\right )+\frac {1}{160} i \coth \left (\frac {x}{2}\right ) \text {csch}^4\left (\frac {x}{2}\right )-\frac {3}{8} \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {3}{8} \log \left (\sinh \left (\frac {x}{2}\right )\right )-\frac {5}{32} \text {sech}^2\left (\frac {x}{2}\right )+\frac {1}{64} \text {sech}^4\left (\frac {x}{2}\right )+\frac {1}{10} i \tanh \left (\frac {x}{2}\right )-\frac {7}{160} i \text {sech}^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right )+\frac {1}{160} i \text {sech}^4\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (27 ) = 54\).
Time = 25.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.81
method | result | size |
risch | \(-\frac {-40 i {\mathrm e}^{8 x}+25 \,{\mathrm e}^{9 x}-10 \,{\mathrm e}^{7 x}-80 i {\mathrm e}^{4 x}+10 \,{\mathrm e}^{3 x}-8 i-25 \,{\mathrm e}^{x}}{20 \left ({\mathrm e}^{2 x}-1\right )^{5}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8}\) | \(65\) |
default | \(\frac {i \tanh \left (\frac {x}{2}\right )}{16}+\frac {i \tanh \left (\frac {x}{2}\right )^{5}}{160}+\frac {\tanh \left (\frac {x}{2}\right )^{4}}{64}+\frac {i \tanh \left (\frac {x}{2}\right )^{3}}{32}+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {i}{160 \tanh \left (\frac {x}{2}\right )^{5}}+\frac {i}{16 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{32 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8}\) | \(93\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.00 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=-\frac {15 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} + 1\right ) - 15 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} - 1\right ) + 50 \, e^{\left (9 \, x\right )} - 80 i \, e^{\left (8 \, x\right )} - 20 \, e^{\left (7 \, x\right )} - 160 i \, e^{\left (4 \, x\right )} + 20 \, e^{\left (3 \, x\right )} - 50 \, e^{x} - 16 i}{40 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (36) = 72\).
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.78 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {3 \log {\left (e^{x} - 1 \right )}}{8} - \frac {3 \log {\left (e^{x} + 1 \right )}}{8} + \frac {- 25 e^{9 x} + 40 i e^{8 x} + 10 e^{7 x} + 80 i e^{4 x} - 10 e^{3 x} + 25 e^{x} + 8 i}{20 e^{10 x} - 100 e^{8 x} + 200 e^{6 x} - 200 e^{4 x} + 100 e^{2 x} - 20} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.53 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {25 \, e^{\left (-x\right )} - 10 \, e^{\left (-3 \, x\right )} - 80 i \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-7 \, x\right )} - 40 i \, e^{\left (-8 \, x\right )} - 25 \, e^{\left (-9 \, x\right )} - 8 i}{20 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} - \frac {3}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=-\frac {25 \, e^{\left (9 \, x\right )} - 40 i \, e^{\left (8 \, x\right )} - 10 \, e^{\left (7 \, x\right )} - 80 i \, e^{\left (4 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 25 \, e^{x} - 8 i}{20 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} - \frac {3}{8} \, \log \left (e^{x} + 1\right ) + \frac {3}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 1.69 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.44 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {3\,\ln \left (\frac {3}{4}-\frac {3\,{\mathrm {e}}^x}{4}\right )}{8}-\frac {3\,\ln \left (\frac {3\,{\mathrm {e}}^x}{4}+\frac {3}{4}\right )}{8}-\frac {5\,{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {9\,{\mathrm {e}}^x}{2\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {6\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^3}-\frac {4\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^4}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {8{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {16{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^3}+\frac {16{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^4}+\frac {32{}\mathrm {i}}{5\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5} \]
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