Integrand size = 13, antiderivative size = 30 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x)-\frac {\text {csch}^3(x)}{3}-i \log (\sinh (x)) \]
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Time = 0.03 (sec) , antiderivative size = 30, 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, 76} \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-\frac {1}{3} \text {csch}^3(x)+\frac {1}{2} i \text {csch}^2(x)-\text {csch}(x)-i \log (\sinh (x)) \]
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Rule 76
Rule 3964
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {(i-i x)^2 (i+i x)}{x^4} \, dx,x,i \sinh (x)\right ) \\ & = \text {Subst}\left (\int \left (-\frac {i}{x^4}+\frac {i}{x^3}+\frac {i}{x^2}-\frac {i}{x}\right ) \, dx,x,i \sinh (x)\right ) \\ & = -\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x)-\frac {\text {csch}^3(x)}{3}-i \log (\sinh (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-\text {csch}(x)+\frac {1}{2} i \text {csch}^2(x)-\frac {\text {csch}^3(x)}{3}-i \log (\sinh (x)) \]
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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. \(2 (24 ) = 48\).
Time = 6.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80
method | result | size |
risch | \(i x -\frac {2 \,{\mathrm e}^{x} \left (-3 i {\mathrm e}^{3 x}+3 \,{\mathrm e}^{4 x}+3 i {\mathrm e}^{x}-2 \,{\mathrm e}^{2 x}+3\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3}}-i \ln \left ({\mathrm e}^{2 x}-1\right )\) | \(54\) |
default | \(\frac {3 \tanh \left (\frac {x}{2}\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )^{3}}{24}+\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}-i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {3}{8 \tanh \left (\frac {x}{2}\right )}+i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(78\) |
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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. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.33 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=\frac {3 i \, x e^{\left (6 \, x\right )} - 3 \, {\left (3 i \, x - 2 i\right )} e^{\left (4 \, x\right )} - 3 \, {\left (-3 i \, x + 2 i\right )} e^{\left (2 \, x\right )} - 3 \, {\left (i \, e^{\left (6 \, x\right )} - 3 i \, e^{\left (4 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 3 i \, x - 6 \, e^{\left (5 \, x\right )} + 4 \, e^{\left (3 \, x\right )} - 6 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} - 1\right )}} \]
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\[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=\int \frac {\coth ^{5}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.50 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=-i \, x + \frac {2 \, {\left (3 \, e^{\left (-x\right )} - 3 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + 3 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=\frac {11 i \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 12 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 i \, e^{\left (-x\right )} - 12 i \, e^{x} + 16}{6 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} - i \, \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right ) \]
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Time = 2.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {\coth ^5(x)}{i+\text {csch}(x)} \, dx=x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,1{}\mathrm {i}-\frac {8\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {\frac {8\,{\mathrm {e}}^x}{3}-2{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {2\,{\mathrm {e}}^x-2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]
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