Integrand size = 14, antiderivative size = 59 \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=-\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))} \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2729, 2727} \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=-\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))}-\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))^2} \]
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Rule 2727
Rule 2729
Rubi steps \begin{align*} \text {integral}& = -\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))^2}+\frac {1}{3} \int \frac {1}{1-i \sinh (c+d x)} \, dx \\ & = -\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=-\frac {\cosh \left (\frac {3}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )}{3 d \left (i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Time = 0.96 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.47
method | result | size |
risch | \(\frac {\frac {2 i}{3}+2 \,{\mathrm e}^{d x +c}}{d \left ({\mathrm e}^{d x +c}+i\right )^{3}}\) | \(28\) |
derivativedivides | \(\frac {-\frac {4}{3 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2}{i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) | \(55\) |
default | \(\frac {-\frac {4}{3 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2}{i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) | \(55\) |
parallelrisch | \(\frac {6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4}{3 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+3 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}\) | \(74\) |
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Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=\frac {2 \, {\left (3 \, e^{\left (d x + c\right )} + i\right )}}{3 \, {\left (d e^{\left (3 \, d x + 3 \, c\right )} + 3 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 3 \, d e^{\left (d x + c\right )} - i \, d\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=\frac {6 e^{c} e^{d x} + 2 i}{3 d e^{3 c} e^{3 d x} + 9 i d e^{2 c} e^{2 d x} - 9 d e^{c} e^{d x} - 3 i d} \]
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Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=\frac {2 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-d x - c\right )} + 3 i \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-3 \, d x - 3 \, c\right )} - i\right )}} - \frac {2 i}{3 \, d {\left (3 \, e^{\left (-d x - c\right )} + 3 i \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-3 \, d x - 3 \, c\right )} - i\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.42 \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=\frac {2 \, {\left (3 \, e^{\left (d x + c\right )} + i\right )}}{3 \, d {\left (e^{\left (d x + c\right )} + i\right )}^{3}} \]
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Time = 1.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx=-\frac {2\,\left (-1+{\mathrm {e}}^{c+d\,x}\,3{}\mathrm {i}\right )}{3\,d\,{\left (-1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^3} \]
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