Optimal. Leaf size=23 \[ -\frac{i \log (-\sinh (c+d x)+i)}{a d} \]
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Rubi [A] time = 0.0273148, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2667, 31} \[ -\frac{i \log (-\sinh (c+d x)+i)}{a d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,i a \sinh (c+d x)\right )}{a d}\\ &=-\frac{i \log (i-\sinh (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0145465, size = 23, normalized size = 1. \[ -\frac{i \log (-\sinh (c+d x)+i)}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 23, normalized size = 1. \begin{align*}{\frac{-i\ln \left ( a+ia\sinh \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12375, size = 27, normalized size = 1.17 \begin{align*} -\frac{i \, \log \left (i \, a \sinh \left (d x + c\right ) + a\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18425, size = 57, normalized size = 2.48 \begin{align*} \frac{i \, d x - 2 i \, \log \left (e^{\left (d x + c\right )} - i\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.53724, size = 22, normalized size = 0.96 \begin{align*} \frac{i x}{a} - \frac{2 i \log{\left (e^{d x} - i e^{- c} \right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23608, size = 45, normalized size = 1.96 \begin{align*} \frac{i \,{\left (d x + c\right )}}{a d} - \frac{2 i \, \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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