Integrand size = 13, antiderivative size = 31 \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=-\text {sech}(x)+\frac {2 \text {sech}^3(x)}{3}-\frac {\text {sech}^5(x)}{5}-\frac {1}{5} i \tanh ^5(x) \]
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Time = 0.05 (sec) , antiderivative size = 31, 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, 2686, 200} \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=-\frac {1}{5} i \tanh ^5(x)-\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]
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Rule 30
Rule 200
Rule 2686
Rule 2687
Rule 2785
Rubi steps \begin{align*} \text {integral}& = -\left (i \int \text {sech}^2(x) \tanh ^4(x) \, dx\right )+\int \text {sech}(x) \tanh ^5(x) \, dx \\ & = -\text {Subst}\left (\int x^4 \, dx,x,i \tanh (x)\right )-\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\text {sech}(x)\right ) \\ & = -\frac {1}{5} i \tanh ^5(x)-\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\text {sech}(x)\right ) \\ & = -\text {sech}(x)+\frac {2 \text {sech}^3(x)}{3}-\frac {\text {sech}^5(x)}{5}-\frac {1}{5} i \tanh ^5(x) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(31)=62\).
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.10 \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=-\frac {200-534 \cosh (x)+288 \cosh (2 x)-178 \cosh (3 x)+24 \cosh (4 x)+64 i \sinh (x)+178 i \sinh (2 x)-192 i \sinh (3 x)+89 i \sinh (4 x)}{960 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^5 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \]
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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. \(2 (24 ) = 48\).
Time = 13.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03
method | result | size |
risch | \(-\frac {2 \left (25 i {\mathrm e}^{4 x}+5 \,{\mathrm e}^{5 x}+21 i {\mathrm e}^{2 x}+13 \,{\mathrm e}^{3 x}+15 i {\mathrm e}^{6 x}+15 \,{\mathrm e}^{7 x}-9 \,{\mathrm e}^{x}+3 i\right )}{15 \left ({\mathrm e}^{x}+i\right )^{5} \left ({\mathrm e}^{x}-i\right )^{3}}\) | \(63\) |
default | \(\frac {3 i}{8 \left (-i+\tanh \left (\frac {x}{2}\right )\right )}+\frac {i}{6 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{3}}+\frac {1}{4 \left (-i+\tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {2 i}{5 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {3 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}\) | \(93\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.77 \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=-\frac {2 \, {\left (15 \, e^{\left (7 \, x\right )} + 15 i \, e^{\left (6 \, x\right )} + 5 \, e^{\left (5 \, x\right )} + 25 i \, e^{\left (4 \, x\right )} + 13 \, e^{\left (3 \, x\right )} + 21 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} + 3 i\right )}}{15 \, {\left (e^{\left (8 \, x\right )} + 2 i \, e^{\left (7 \, x\right )} + 2 \, e^{\left (6 \, x\right )} + 6 i \, e^{\left (5 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (26) = 52\).
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.45 \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=\frac {- 30 e^{7 x} - 30 i e^{6 x} - 10 e^{5 x} - 50 i e^{4 x} - 26 e^{3 x} - 42 i e^{2 x} + 18 e^{x} - 6 i}{15 e^{8 x} + 30 i e^{7 x} + 30 e^{6 x} + 90 i e^{5 x} + 90 i e^{3 x} - 30 e^{2 x} + 30 i e^{x} - 15} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (23) = 46\).
Time = 0.23 (sec) , antiderivative size = 413, normalized size of antiderivative = 13.32 \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=\frac {18 \, e^{\left (-x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {42 i \, e^{\left (-2 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {26 \, e^{\left (-3 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {50 i \, e^{\left (-4 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {10 \, e^{\left (-5 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {30 i \, e^{\left (-6 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac {30 \, e^{\left (-7 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac {6 i}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=-\frac {15 \, e^{\left (2 \, x\right )} - 24 i \, e^{x} - 13}{24 \, {\left (e^{x} - i\right )}^{3}} - \frac {165 \, e^{\left (4 \, x\right )} + 480 i \, e^{\left (3 \, x\right )} - 650 \, e^{\left (2 \, x\right )} - 400 i \, e^{x} + 113}{120 \, {\left (e^{x} + i\right )}^{5}} \]
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Time = 1.92 (sec) , antiderivative size = 231, normalized size of antiderivative = 7.45 \[ \int \frac {\tanh ^4(x)}{i+\sinh (x)} \, dx=-\frac {1}{6\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}\right )}-\frac {\frac {11\,{\mathrm {e}}^x}{40}+\frac {1}{8}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {\frac {11\,{\mathrm {e}}^{2\,x}}{40}-\frac {17}{120}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{4}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}+\frac {1{}\mathrm {i}}{4\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}-\frac {5}{8\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}-\frac {11}{40\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}-\frac {\frac {{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}}{8}+\frac {11\,{\mathrm {e}}^{3\,x}}{40}-\frac {17\,{\mathrm {e}}^x}{40}-\frac {1}{8}{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {\frac {11\,{\mathrm {e}}^{4\,x}}{40}-\frac {17\,{\mathrm {e}}^{2\,x}}{20}+\frac {11}{40}+\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}}{{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+5\,{\mathrm {e}}^x+1{}\mathrm {i}} \]
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