Integrand size = 13, antiderivative size = 32 \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=x+\frac {i \cosh (x)}{3 (i+\sinh (x))^2}-\frac {5 \cosh (x)}{3 (i+\sinh (x))} \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2837, 2814, 2727} \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=x-\frac {5 \cosh (x)}{3 (\sinh (x)+i)}+\frac {i \cosh (x)}{3 (\sinh (x)+i)^2} \]
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Rule 2727
Rule 2814
Rule 2837
Rubi steps \begin{align*} \text {integral}& = \frac {i \cosh (x)}{3 (i+\sinh (x))^2}+\frac {1}{3} \int \frac {-2 i+3 \sinh (x)}{i+\sinh (x)} \, dx \\ & = x+\frac {i \cosh (x)}{3 (i+\sinh (x))^2}-\frac {5}{3} i \int \frac {1}{i+\sinh (x)} \, dx \\ & = x+\frac {i \cosh (x)}{3 (i+\sinh (x))^2}-\frac {5 \cosh (x)}{3 (i+\sinh (x))} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{3} i \cosh (x) \left (-\frac {6 \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {\cosh ^2(x)}}+\frac {4-5 i \sinh (x)}{(i+\sinh (x))^2}\right ) \]
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Time = 1.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81
method | result | size |
risch | \(x +\frac {2 i \left (9 i {\mathrm e}^{x}+6 \,{\mathrm e}^{2 x}-5\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}\) | \(26\) |
default | \(-\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {2}{\tanh \left (\frac {x}{2}\right )+i}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(52\) |
parallelrisch | \(\frac {\left (-3 \tanh \left (\frac {x}{2}\right )^{3}-9 i \tanh \left (\frac {x}{2}\right )^{2}+9 \tanh \left (\frac {x}{2}\right )+3 i\right ) \ln \left (1-\tanh \left (\frac {x}{2}\right )\right )+\left (3 \tanh \left (\frac {x}{2}\right )^{3}+9 i \tanh \left (\frac {x}{2}\right )^{2}-9 \tanh \left (\frac {x}{2}\right )-3 i\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-2 i \tanh \left (\frac {x}{2}\right )^{3}-12 i \tanh \left (\frac {x}{2}\right )+6}{3 \tanh \left (\frac {x}{2}\right )^{3}+9 i \tanh \left (\frac {x}{2}\right )^{2}-9 \tanh \left (\frac {x}{2}\right )-3 i}\) | \(118\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=\frac {3 \, x e^{\left (3 \, x\right )} - 3 \, {\left (-3 i \, x - 4 i\right )} e^{\left (2 \, x\right )} - 9 \, {\left (x + 2\right )} e^{x} - 3 i \, x - 10 i}{3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=x + \frac {12 i e^{2 x} - 18 e^{x} - 10 i}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} \]
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none
Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=x - \frac {2 \, {\left (9 \, e^{\left (-x\right )} + 6 i \, e^{\left (-2 \, x\right )} - 5 i\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i\right )}} \]
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none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=x - \frac {2 \, {\left (-6 i \, e^{\left (2 \, x\right )} + 9 \, e^{x} + 5 i\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} \]
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Time = 1.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx=x+\frac {-\frac {2}{3}+\frac {{\mathrm {e}}^x\,4{}\mathrm {i}}{3}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {\frac {4\,{\mathrm {e}}^x}{3}-\frac {{\mathrm {e}}^{2\,x}\,4{}\mathrm {i}}{3}+\frac {4}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}+\frac {4{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]
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