Integrand size = 13, antiderivative size = 77 \[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {11}{16} i \log (i-\sinh (x))-\frac {5}{16} i \log (i+\sinh (x))-\frac {i}{8 (1-i \sinh (x))}+\frac {i}{8 (1+i \sinh (x))^2}-\frac {3 i}{4 (1+i \sinh (x))} \]
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Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3964, 90} \[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {i}{8 (1-i \sinh (x))}-\frac {3 i}{4 (1+i \sinh (x))}+\frac {i}{8 (1+i \sinh (x))^2}-\frac {11}{16} i \log (-\sinh (x)+i)-\frac {5}{16} i \log (\sinh (x)+i) \]
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Rule 90
Rule 3964
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^4}{(i-i x)^2 (i+i x)^3} \, dx,x,i \sinh (x)\right ) \\ & = \text {Subst}\left (\int \left (-\frac {i}{8 (-1+x)^2}-\frac {5 i}{16 (-1+x)}-\frac {i}{4 (1+x)^3}+\frac {3 i}{4 (1+x)^2}-\frac {11 i}{16 (1+x)}\right ) \, dx,x,i \sinh (x)\right ) \\ & = -\frac {11}{16} i \log (i-\sinh (x))-\frac {5}{16} i \log (i+\sinh (x))-\frac {i}{8 (1-i \sinh (x))}+\frac {i}{8 (1+i \sinh (x))^2}-\frac {3 i}{4 (1+i \sinh (x))} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{16} \left (-11 i \log (i-\sinh (x))-5 i \log (i+\sinh (x))-\frac {2 \left (6+3 i \sinh (x)+5 \sinh ^2(x)\right )}{(-i+\sinh (x))^2 (i+\sinh (x))}\right ) \]
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Time = 1.67 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92
method | result | size |
risch | \(i x -\frac {5 \,{\mathrm e}^{x}-6 i {\mathrm e}^{2 x}+14 \,{\mathrm e}^{3 x}+6 i {\mathrm e}^{4 x}+5 \,{\mathrm e}^{5 x}}{4 \left ({\mathrm e}^{x}+i\right )^{2} \left ({\mathrm e}^{x}-i\right )^{4}}-\frac {5 i \ln \left ({\mathrm e}^{x}+i\right )}{8}-\frac {11 i \ln \left ({\mathrm e}^{x}-i\right )}{8}\) | \(71\) |
default | \(i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {5 i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{8}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {11 i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{8}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {1}{\tanh \left (\frac {x}{2}\right )-i}\) | \(109\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (47) = 94\).
Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.42 \[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {8 i \, x e^{\left (6 \, x\right )} + 2 \, {\left (8 \, x - 5\right )} e^{\left (5 \, x\right )} - 4 \, {\left (-2 i \, x + 3 i\right )} e^{\left (4 \, x\right )} + 4 \, {\left (8 \, x - 7\right )} e^{\left (3 \, x\right )} - 4 \, {\left (2 i \, x - 3 i\right )} e^{\left (2 \, x\right )} + 2 \, {\left (8 \, x - 5\right )} e^{x} - 5 \, {\left (i \, e^{\left (6 \, x\right )} + 2 \, e^{\left (5 \, x\right )} + i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - i\right )} \log \left (e^{x} + i\right ) - 11 \, {\left (i \, e^{\left (6 \, x\right )} + 2 \, e^{\left (5 \, x\right )} + i \, e^{\left (4 \, x\right )} + 4 \, e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - i\right )} \log \left (e^{x} - i\right ) - 8 i \, x}{8 \, {\left (e^{\left (6 \, x\right )} - 2 i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )}} \]
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\[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\tanh ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (47) = 94\).
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.25 \[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=-i \, x + \frac {5 \, e^{\left (-x\right )} + 6 i \, e^{\left (-2 \, x\right )} + 14 \, e^{\left (-3 \, x\right )} - 6 i \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )}}{8 i \, e^{\left (-x\right )} - 4 \, e^{\left (-2 \, x\right )} + 16 i \, e^{\left (-3 \, x\right )} + 4 \, e^{\left (-4 \, x\right )} + 8 i \, e^{\left (-5 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - 4} - \frac {5}{8} i \, \log \left (e^{\left (-x\right )} - i\right ) - \frac {11}{8} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22 \[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {5 \, e^{\left (-x\right )} - 5 \, e^{x} - 6 i}{16 \, {\left (-i \, e^{\left (-x\right )} + i \, e^{x} - 2\right )}} + \frac {33 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 84 \, e^{\left (-x\right )} + 84 \, e^{x} - 52 i}{32 \, {\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}^{2}} - \frac {5}{16} i \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {11}{16} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
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Time = 0.62 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.82 \[ \int \frac {\tanh ^3(x)}{i+\text {csch}(x)} \, dx=x\,1{}\mathrm {i}-\ln \left (\left (\frac {3\,{\mathrm {e}}^x}{4}-\frac {3}{4}{}\mathrm {i}\right )\,\left (\frac {3\,{\mathrm {e}}^x}{4}+\frac {3}{4}{}\mathrm {i}\right )\right )\,1{}\mathrm {i}+\frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}-\frac {1}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {1{}\mathrm {i}}{2\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}\right )}+\frac {2{}\mathrm {i}}{1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {3}{2\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}+\frac {1}{4\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]
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