Integrand size = 24, antiderivative size = 52 \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {x}{a}-\frac {i \cosh (c+d x)}{a d}-\frac {i \cosh (c+d x)}{a d (1+i \sinh (c+d x))} \]
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Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2825, 12, 2814, 2727} \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i \cosh (c+d x)}{a d}-\frac {i \cosh (c+d x)}{a d (1+i \sinh (c+d x))}+\frac {x}{a} \]
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Rule 12
Rule 2727
Rule 2814
Rule 2825
Rubi steps \begin{align*} \text {integral}& = -\frac {i \cosh (c+d x)}{a d}+\frac {i \int \frac {a \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx}{a} \\ & = -\frac {i \cosh (c+d x)}{a d}+i \int \frac {\sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {x}{a}-\frac {i \cosh (c+d x)}{a d}-\int \frac {1}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {x}{a}-\frac {i \cosh (c+d x)}{a d}-\frac {i \cosh (c+d x)}{d (a+i a \sinh (c+d x))} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\cosh (c+d x) \left (\frac {\text {arcsinh}(\sinh (c+d x))}{\sqrt {\cosh ^2(c+d x)}}+\frac {-2-i \sinh (c+d x)}{-i+\sinh (c+d x)}\right )}{a d} \]
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Time = 1.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {x}{a}-\frac {i {\mathrm e}^{d x +c}}{2 a d}-\frac {i {\mathrm e}^{-d x -c}}{2 a d}-\frac {2 i}{d a \left ({\mathrm e}^{d x +c}-i\right )}\) | \(60\) |
derivativedivides | \(\frac {-\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {i}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {8 i}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-8}-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}\) | \(86\) |
default | \(\frac {-\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {i}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {8 i}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-8}-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}\) | \(86\) |
parallelrisch | \(\frac {\left (-2 d x -3 i\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i d x \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+i \sinh \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+\cosh \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )}{2 a d \left (i \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(99\) |
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Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.33 \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (2 \, d x - 1\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-2 i \, d x - 5 i\right )} e^{\left (d x + c\right )} - i \, e^{\left (3 \, d x + 3 \, c\right )} - 1}{2 \, {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.90 \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\begin {cases} \frac {\left (- 2 i a d e^{2 c} e^{d x} - 2 i a d e^{- d x}\right ) e^{- c}}{4 a^{2} d^{2}} & \text {for}\: a^{2} d^{2} e^{c} \neq 0 \\x \left (\frac {\left (- i e^{2 c} + 2 e^{c} + i\right ) e^{- c}}{2 a} - \frac {1}{a}\right ) & \text {otherwise} \end {cases} - \frac {2 i}{a d e^{c} e^{d x} - i a d} + \frac {x}{a} \]
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Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.42 \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {d x + c}{a d} + \frac {-5 i \, e^{\left (-d x - c\right )} + 1}{2 \, {\left (i \, a e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} - \frac {i \, e^{\left (-d x - c\right )}}{2 \, a d} \]
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Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\frac {2 \, {\left (d x + c\right )}}{a} - \frac {i \, e^{\left (d x + c\right )}}{a} - \frac {{\left (5 i \, e^{\left (d x + c\right )} + 1\right )} e^{\left (-d x - c\right )}}{a {\left (e^{\left (d x + c\right )} - i\right )}}}{2 \, d} \]
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Time = 1.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {x}{a}-\frac {2{}\mathrm {i}}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}}{2\,a\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,1{}\mathrm {i}}{2\,a\,d} \]
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