Integrand size = 13, antiderivative size = 14 \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=x-\frac {2 \cosh (x)}{i+\sinh (x)} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2759, 8} \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=x-\frac {2 \cosh (x)}{\sinh (x)+i} \]
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Rule 8
Rule 2759
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cosh (x)}{i+\sinh (x)}+\int 1 \, dx \\ & = x-\frac {2 \cosh (x)}{i+\sinh (x)} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(14)=28\).
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 4.93 \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=\frac {2 \cosh ^3(x) \left (-1-\frac {\arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \sqrt {1-i \sinh (x)}}{\sqrt {1+i \sinh (x)}}\right )}{(-i+\sinh (x)) (i+\sinh (x))^2} \]
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Time = 13.10 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
risch | \(x +\frac {4 i}{{\mathrm e}^{x}+i}\) | \(13\) |
default | \(-\frac {4}{\tanh \left (\frac {x}{2}\right )+i}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(29\) |
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none
Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=\frac {x e^{x} + i \, x + 4 i}{e^{x} + i} \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=x + \frac {4 i}{e^{x} + i} \]
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none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=x + \frac {4 i}{e^{\left (-x\right )} - i} \]
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=x + \frac {4 i}{e^{x} + i} \]
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Time = 1.32 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh ^2(x)}{(i+\sinh (x))^2} \, dx=x+\frac {4{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
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