Optimal. Leaf size=34 \[ \frac{\sinh (c+d x)}{a d}-\frac{i \sinh ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.047608, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {2667} \[ \frac{\sinh (c+d x)}{a d}-\frac{i \sinh ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2667
Rubi steps
\begin{align*} \int \frac{\cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \operatorname{Subst}(\int (a-x) \, dx,x,i a \sinh (c+d x))}{a^3 d}\\ &=\frac{\sinh (c+d x)}{a d}-\frac{i \sinh ^2(c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.0491364, size = 28, normalized size = 0.82 \[ \frac{(2-i \sinh (c+d x)) \sinh (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 29, normalized size = 0.9 \begin{align*} -{\frac{{\frac{i}{2}} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}-\sinh \left ( dx+c \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16329, size = 81, normalized size = 2.38 \begin{align*} -\frac{i \,{\left (4 i \, e^{\left (-d x - c\right )} + 1\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} - \frac{i \,{\left (-4 i \, e^{\left (-d x - c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{8 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08925, size = 120, normalized size = 3.53 \begin{align*} \frac{{\left (-i \, e^{\left (4 \, d x + 4 \, c\right )} + 4 \, e^{\left (3 \, d x + 3 \, c\right )} - 4 \, e^{\left (d x + c\right )} - i\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.766, size = 136, normalized size = 4. \begin{align*} \begin{cases} \frac{\left (- 32 i a^{3} d^{3} e^{5 c} e^{2 d x} + 128 a^{3} d^{3} e^{4 c} e^{d x} - 128 a^{3} d^{3} e^{2 c} e^{- d x} - 32 i a^{3} d^{3} e^{c} e^{- 2 d x}\right ) e^{- 3 c}}{256 a^{4} d^{4}} & \text{for}\: 256 a^{4} d^{4} e^{3 c} \neq 0 \\- \frac{x \left (i e^{4 c} - 2 e^{3 c} - 2 e^{c} - i\right ) e^{- 2 c}}{4 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15882, size = 65, normalized size = 1.91 \begin{align*} -\frac{{\left (4 \, e^{\left (d x + c\right )} + i\right )} e^{\left (-2 \, d x - 2 \, c\right )} + i \, e^{\left (2 \, d x + 2 \, c\right )} - 4 \, e^{\left (d x + c\right )}}{8 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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