Optimal. Leaf size=88 \[ -\frac {2 i \cosh (c+d x)}{15 d (1-i \sinh (c+d x))}-\frac {2 i \cosh (c+d x)}{15 d (1-i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{5 d (1-i \sinh (c+d x))^3} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2650, 2648} \[ -\frac {2 i \cosh (c+d x)}{15 d (1-i \sinh (c+d x))}-\frac {2 i \cosh (c+d x)}{15 d (1-i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{5 d (1-i \sinh (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rubi steps
\begin {align*} \int \frac {1}{(1-i \sinh (c+d x))^3} \, dx &=-\frac {i \cosh (c+d x)}{5 d (1-i \sinh (c+d x))^3}+\frac {2}{5} \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx\\ &=-\frac {i \cosh (c+d x)}{5 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{15 d (1-i \sinh (c+d x))^2}+\frac {2}{15} \int \frac {1}{1-i \sinh (c+d x)} \, dx\\ &=-\frac {i \cosh (c+d x)}{5 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{15 d (1-i \sinh (c+d x))^2}-\frac {2 i \cosh (c+d x)}{15 d (1-i \sinh (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 81, normalized size = 0.92 \[ \frac {-15 i \sinh (c+d x)+6 i \sinh (2 (c+d x))+i \sinh (3 (c+d x))-15 \cosh (c+d x)-6 \cosh (2 (c+d x))+\cosh (3 (c+d x))+10}{30 d (\sinh (c+d x)+i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 85, normalized size = 0.97 \[ \frac {40 i \, e^{\left (2 \, d x + 2 \, c\right )} - 20 \, e^{\left (d x + c\right )} - 4 i}{15 \, d e^{\left (5 \, d x + 5 \, c\right )} + 75 i \, d e^{\left (4 \, d x + 4 \, c\right )} - 150 \, d e^{\left (3 \, d x + 3 \, c\right )} - 150 i \, d e^{\left (2 \, d x + 2 \, c\right )} + 75 \, d e^{\left (d x + c\right )} + 15 i \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 36, normalized size = 0.41 \[ \frac {i \, {\left (40 \, e^{\left (2 \, d x + 2 \, c\right )} + 20 i \, e^{\left (d x + c\right )} - 4\right )}}{15 \, d {\left (e^{\left (d x + c\right )} + i\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 88, normalized size = 1.00 \[ \frac {\frac {8}{5 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}+\frac {4 i}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}-\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}-\frac {4 i}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {2}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 211, normalized size = 2.40 \[ \frac {20 i \, e^{\left (-d x - c\right )}}{d {\left (75 i \, e^{\left (-d x - c\right )} - 150 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 75 \, e^{\left (-4 \, d x - 4 \, c\right )} + 15 i \, e^{\left (-5 \, d x - 5 \, c\right )} + 15\right )}} - \frac {40 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (75 i \, e^{\left (-d x - c\right )} - 150 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 75 \, e^{\left (-4 \, d x - 4 \, c\right )} + 15 i \, e^{\left (-5 \, d x - 5 \, c\right )} + 15\right )}} + \frac {4}{d {\left (75 i \, e^{\left (-d x - c\right )} - 150 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 75 \, e^{\left (-4 \, d x - 4 \, c\right )} + 15 i \, e^{\left (-5 \, d x - 5 \, c\right )} + 15\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 40, normalized size = 0.45 \[ -\frac {4\,\left (10\,{\mathrm {e}}^{2\,c+2\,d\,x}-1+{\mathrm {e}}^{c+d\,x}\,5{}\mathrm {i}\right )}{15\,d\,{\left (-1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 114, normalized size = 1.30 \[ \frac {- 4 e^{5 c} - 20 i e^{4 c} e^{- d x} + 40 e^{3 c} e^{- 2 d x}}{- 15 d e^{5 c} - 75 i d e^{4 c} e^{- d x} + 150 d e^{3 c} e^{- 2 d x} + 150 i d e^{2 c} e^{- 3 d x} - 75 d e^{c} e^{- 4 d x} - 15 i d e^{- 5 d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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