Integrand size = 15, antiderivative size = 23 \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=B x-\frac {(i A+B) \cosh (x)}{i+\sinh (x)} \]
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Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2814, 2727} \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=B x-\frac {(B+i A) \cosh (x)}{\sinh (x)+i} \]
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Rule 2727
Rule 2814
Rubi steps \begin{align*} \text {integral}& = B x-(-A+i B) \int \frac {1}{i+\sinh (x)} \, dx \\ & = B x-\frac {(i A+B) \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. \(53\) vs. \(2(23)=46\).
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.30 \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=\cosh (x) \left (\frac {2 i B \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {\cosh ^2(x)}}-\frac {i A+B}{i+\sinh (x)}\right ) \]
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Time = 0.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13
method | result | size |
risch | \(B x -\frac {2 A}{{\mathrm e}^{x}+i}+\frac {2 i B}{{\mathrm e}^{x}+i}\) | \(26\) |
parallelrisch | \(\frac {i B x +x \tanh \left (\frac {x}{2}\right ) B -2 i A -2 B}{\tanh \left (\frac {x}{2}\right )+i}\) | \(31\) |
default | \(B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2 i \left (-i B +A \right )}{\tanh \left (\frac {x}{2}\right )+i}-B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(39\) |
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=\frac {B x e^{x} + i \, B x - 2 \, A + 2 i \, B}{e^{x} + i} \]
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Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=B x + \frac {- 2 A + 2 i B}{e^{x} + i} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=B {\left (x + \frac {2 i}{e^{\left (-x\right )} - i}\right )} - \frac {2 \, A}{e^{\left (-x\right )} - i} \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=B x - \frac {2 \, {\left (A - i \, B\right )}}{e^{x} + i} \]
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Time = 0.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \sinh (x)}{i+\sinh (x)} \, dx=B\,x-\frac {2\,A-B\,2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
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