Optimal. Leaf size=35 \[ -\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))}-\frac{i x}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0438023, antiderivative size = 35, 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 = {2735, 2648} \[ -\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))}-\frac{i x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i x}{a}+i \int \frac{1}{a+i a \sinh (c+d x)} \, dx\\ &=-\frac{i x}{a}-\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.206753, size = 61, normalized size = 1.74 \[ \frac{i \cosh (c+d x) \left (1-\frac{\sinh ^{-1}(\sinh (c+d x)) (\sinh (c+d x)-i)}{\sqrt{\cosh ^2(c+d x)}}\right )}{a d (\sinh (c+d x)-i)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 67, normalized size = 1.9 \begin{align*}{\frac{-i}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{2\,i}{da} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{i}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.1521, size = 49, normalized size = 1.4 \begin{align*} -\frac{i \,{\left (d x + c\right )}}{a d} - \frac{2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.32153, size = 78, normalized size = 2.23 \begin{align*} \frac{-i \, d x e^{\left (d x + c\right )} - d x - 2}{a d e^{\left (d x + c\right )} - i \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.352585, size = 24, normalized size = 0.69 \begin{align*} - \frac{i x}{a} - \frac{2 e^{- c}}{a d \left (e^{d x} - i e^{- c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23488, size = 46, normalized size = 1.31 \begin{align*} -\frac{i \,{\left (d x + c\right )}}{a d} - \frac{2 i}{a d{\left (i \, e^{\left (d x + c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]