Integrand size = 14, antiderivative size = 117 \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))} \]
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Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2729, 2727} \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4} \]
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Rule 2727
Rule 2729
Rubi steps \begin{align*} \text {integral}& = \frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3}{7} \int \frac {1}{(1+i \sinh (c+d x))^3} \, dx \\ & = \frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {6}{35} \int \frac {1}{(1+i \sinh (c+d x))^2} \, dx \\ & = \frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2}{35} \int \frac {1}{1+i \sinh (c+d x)} \, dx \\ & = \frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=\frac {21 i \cosh \left (\frac {3}{2} (c+d x)\right )-i \cosh \left (\frac {7}{2} (c+d x)\right )+35 \sinh \left (\frac {1}{2} (c+d x)\right )-7 \sinh \left (\frac {5}{2} (c+d x)\right )}{70 d \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^7} \]
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Time = 1.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.44
method | result | size |
risch | \(-\frac {4 \left (-7 \,{\mathrm e}^{d x +c}-21 i {\mathrm e}^{2 d x +2 c}+35 \,{\mathrm e}^{3 d x +3 c}+i\right )}{35 \left ({\mathrm e}^{d x +c}-i\right )^{7} d}\) | \(51\) |
derivativedivides | \(\frac {\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {16 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {72}{5 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {16}{7 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {8 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {12}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {6 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) | \(121\) |
default | \(\frac {\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {16 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {72}{5 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {16}{7 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {8 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {12}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {6 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) | \(121\) |
parallelrisch | \(-\frac {2 \left (6 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+7 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-21 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-21 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+7 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+6\right )}{35 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-7 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-21 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+35 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+35 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-21 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}\) | \(169\) |
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Time = 0.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=-\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} - 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} + i\right )}}{35 \, {\left (d e^{\left (7 \, d x + 7 \, c\right )} - 7 i \, d e^{\left (6 \, d x + 6 \, c\right )} - 21 \, d e^{\left (5 \, d x + 5 \, c\right )} + 35 i \, d e^{\left (4 \, d x + 4 \, c\right )} + 35 \, d e^{\left (3 \, d x + 3 \, c\right )} - 21 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 7 \, d e^{\left (d x + c\right )} + i \, d\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=\frac {- 140 e^{3 c} e^{3 d x} + 84 i e^{2 c} e^{2 d x} + 28 e^{c} e^{d x} - 4 i}{35 d e^{7 c} e^{7 d x} - 245 i d e^{6 c} e^{6 d x} - 735 d e^{5 c} e^{5 d x} + 1225 i d e^{4 c} e^{4 d x} + 1225 d e^{3 c} e^{3 d x} - 735 i d e^{2 c} e^{2 d x} - 245 d e^{c} e^{d x} + 35 i d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (93) = 186\).
Time = 0.19 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.18 \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=\frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} - \frac {12 i \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} - \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} + \frac {4 i}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=-\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} - 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} + i\right )}}{35 \, d {\left (e^{\left (d x + c\right )} - i\right )}^{7}} \]
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Time = 1.89 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx=-\frac {\left (7\,{\mathrm {e}}^{c+d\,x}+{\mathrm {e}}^{2\,c+2\,d\,x}\,21{}\mathrm {i}-35\,{\mathrm {e}}^{3\,c+3\,d\,x}-\mathrm {i}\right )\,4{}\mathrm {i}}{35\,d\,{\left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^7} \]
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