Integrand size = 13, antiderivative size = 8 \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=-i x+\cosh (x) \]
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Time = 0.02 (sec) , antiderivative size = 8, 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 = {2761, 8} \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=\cosh (x)-i x \]
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Rule 8
Rule 2761
Rubi steps \begin{align*} \text {integral}& = \cosh (x)-i \int 1 \, dx \\ & = -i x+\cosh (x) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(34\) vs. \(2(8)=16\).
Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 4.25 \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=\cosh (x)+2 \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \sqrt {\cosh ^2(x)} \text {sech}(x) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7 ) = 14\).
Time = 7.89 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.00
method | result | size |
risch | \(-i x +\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}\) | \(16\) |
default | \(i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}\) | \(40\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.12 \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=\frac {1}{2} \, {\left (-2 i \, x e^{x} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (5) = 10\).
Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=- i x + \frac {e^{x}}{2} + \frac {e^{- x}}{2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=-i \, x + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=-i \, x + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
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Time = 1.18 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx=\mathrm {cosh}\left (x\right )-x\,1{}\mathrm {i} \]
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