Integrand size = 15, antiderivative size = 20 \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=-\frac {i \cosh ^3(x)}{3 (1-i \sinh (x))^3} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2750} \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=-\frac {i \cosh ^3(x)}{3 (1-i \sinh (x))^3} \]
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Rule 2750
Rubi steps \begin{align*} \text {integral}& = -\frac {i \cosh ^3(x)}{3 (1-i \sinh (x))^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=-\frac {\cosh ^3(x)}{3 (i+\sinh (x))^3} \]
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Time = 57.56 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {2 i \left (3 \,{\mathrm e}^{2 x}-1\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}\) | \(19\) |
default | \(\frac {2}{\tanh \left (\frac {x}{2}\right )+i}-\frac {4 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {8}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}\) | \(36\) |
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none
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=-\frac {2 \, {\left (-3 i \, e^{\left (2 \, x\right )} + i\right )}}{3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=\frac {6 i e^{2 x} - 2 i}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} \]
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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. \(2 (14) = 28\).
Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.65 \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=-\frac {6 \, e^{\left (-2 \, x\right )}}{-9 i \, e^{\left (-x\right )} + 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} - 3} + \frac {2}{-9 i \, e^{\left (-x\right )} + 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} - 3} \]
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none
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=-\frac {2 \, {\left (-3 i \, e^{\left (2 \, x\right )} + i\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} \]
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Time = 1.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh ^2(x)}{(1-i \sinh (x))^3} \, dx=\frac {2\,\left (3\,{\mathrm {e}}^{2\,x}-1\right )}{3\,{\left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]
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