Integrand size = 11, antiderivative size = 10 \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{i+\sinh (x)} \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2746, 32} \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{\sinh (x)+i} \]
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Rule 32
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(i+x)^2} \, dx,x,\sinh (x)\right ) \\ & = -\frac {1}{i+\sinh (x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{i+\sinh (x)} \]
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Time = 10.89 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {1}{i+\sinh \left (x \right )}\) | \(10\) |
default | \(-\frac {1}{i+\sinh \left (x \right )}\) | \(10\) |
risch | \(-\frac {2 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+i\right )^{2}}\) | \(12\) |
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, e^{x}}{e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.90 \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=- \frac {2 e^{x}}{e^{2 x} + 2 i e^{x} - 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{\sinh \left (x\right ) + i} \]
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, e^{x}}{{\left (e^{x} + i\right )}^{2}} \]
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Time = 1.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\cosh (x)}{(i+\sinh (x))^2} \, dx=-\frac {1{}\mathrm {i}}{-1+\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}} \]
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