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 = 0.0272801, antiderivative size = 59, normalized size of antiderivative = 1., 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 2650
Rule 2648
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*}
Mathematica [A] time = 0.0710138, size = 59, normalized size = 1. \[ -\frac{\cosh \left (\frac{3}{2} (c+d x)\right )+3 i \sinh \left (\frac{1}{2} (c+d x)\right )}{3 d \left (\sinh \left (\frac{1}{2} (c+d x)\right )+i \cosh \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 55, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({-2\,i \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) ^{-2}}-{\frac{4}{3} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) ^{-3}}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10345, size = 127, normalized size = 2.15 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93858, size = 128, normalized size = 2.17 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.935036, size = 63, normalized size = 1.07 \begin{align*} \frac{\frac{2 e^{- 2 c} e^{d x}}{d} + \frac{2 i e^{- 3 c}}{3 d}}{e^{3 d x} + 3 i e^{- c} e^{2 d x} - 3 e^{- 2 c} e^{d x} - i e^{- 3 c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36704, size = 34, normalized size = 0.58 \begin{align*} \frac{6 \, e^{\left (d x + c\right )} + 2 i}{3 \, d{\left (e^{\left (d x + c\right )} + i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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