Optimal. Leaf size=59 \[ -\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))}-\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))^2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2650, 2648} \[ -\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))}-\frac {i \cosh (c+d x)}{3 d (1-i \sinh (c+d x))^2} \]
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))^2} \, dx &=-\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*}
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Mathematica [A] time = 0.08, size = 59, normalized size = 1.00 \[ -\frac {\cosh \left (\frac {3}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\sinh \left (\frac {1}{2} (c+d x)\right )+i \cosh \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 50, normalized size = 0.85 \[ \frac {6 \, e^{\left (d x + c\right )} + 2 i}{3 \, d e^{\left (3 \, d x + 3 \, c\right )} + 9 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 9 \, d e^{\left (d x + c\right )} - 3 i \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 25, normalized size = 0.42 \[ \frac {6 \, e^{\left (d x + c\right )} + 2 i}{3 \, d {\left (e^{\left (d x + c\right )} + i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 55, normalized size = 0.93 \[ \frac {-\frac {2 i}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {2}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i}-\frac {4}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 94, normalized size = 1.59 \[ \frac {6 \, e^{\left (-d x - c\right )}}{d {\left (9 \, e^{\left (-d x - c\right )} + 9 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} - 3 i\right )}} - \frac {2 i}{d {\left (9 \, e^{\left (-d x - c\right )} + 9 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} - 3 i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 29, normalized size = 0.49 \[ -\frac {2\,\left (-1+{\mathrm {e}}^{c+d\,x}\,3{}\mathrm {i}\right )}{3\,d\,{\left (-1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 68, normalized size = 1.15 \[ \frac {2 e^{3 c} + 6 i e^{2 c} e^{- d x}}{3 d e^{3 c} + 9 i d e^{2 c} e^{- d x} - 9 d e^{c} e^{- 2 d x} - 3 i d e^{- 3 d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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