Integrand size = 11, antiderivative size = 7 \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=\log (i+\sinh (x)) \]
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Time = 0.01 (sec) , antiderivative size = 7, 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, 31} \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=\log (\sinh (x)+i) \]
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Rule 31
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{i+x} \, dx,x,\sinh (x)\right ) \\ & = \log (i+\sinh (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=\log (i+\sinh (x)) \]
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Time = 3.33 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\ln \left (i+\sinh \left (x \right )\right )\) | \(7\) |
default | \(\ln \left (i+\sinh \left (x \right )\right )\) | \(7\) |
risch | \(-x +2 \ln \left ({\mathrm e}^{x}+i\right )\) | \(13\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (5) = 10\).
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.57 \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=-x + 2 \, \log \left (e^{x} + i\right ) \]
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Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.14 \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=- x + 2 \log {\left (e^{x} + i \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=\log \left (\sinh \left (x\right ) + i\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (5) = 10\).
Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.57 \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=-x + 2 \, \log \left (e^{x} + i\right ) \]
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Time = 1.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.43 \[ \int \frac {\cosh (x)}{i+\sinh (x)} \, dx=\ln \left (\mathrm {cosh}\left (x\right )\right )-\mathrm {atan}\left (\mathrm {sinh}\left (x\right )\right )\,1{}\mathrm {i} \]
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