Optimal. Leaf size=27 \[ \frac{i \cosh (c+d x)}{d (1+i \sinh (c+d x))} \]
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Rubi [A] time = 0.0113079, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2648} \[ \frac{i \cosh (c+d x)}{d (1+i \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{1+i \sinh (c+d x)} \, dx &=\frac{i \cosh (c+d x)}{d (1+i \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0608458, size = 42, normalized size = 1.56 \[ \frac{2 \sinh \left (\frac{1}{2} (c+d x)\right )}{d \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 20, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{d \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18467, size = 27, normalized size = 1. \begin{align*} -\frac{2}{d{\left (i \, e^{\left (-d x - c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96137, size = 36, normalized size = 1.33 \begin{align*} \frac{2 i}{d e^{\left (d x + c\right )} - i \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.273093, size = 19, normalized size = 0.7 \begin{align*} \frac{2 i e^{c}}{d \left (i e^{c} + e^{- d x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3587, size = 20, normalized size = 0.74 \begin{align*} \frac{2 i}{d{\left (e^{\left (d x + c\right )} - i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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