Optimal. Leaf size=37 \[ \frac{x}{4}-\frac{i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0132772, antiderivative size = 37, 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 = {2657} \[ \frac{x}{4}-\frac{i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2657
Rubi steps
\begin{align*} \int \frac{1}{5+3 i \sinh (c+d x)} \, dx &=\frac{x}{4}-\frac{i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d}\\ \end{align*}
Mathematica [B] time = 0.0335293, size = 171, normalized size = 4.62 \[ -\frac{\log (5 \cosh (c+d x)-4 \sinh (c+d x))}{8 d}+\frac{\log (4 \sinh (c+d x)+5 \cosh (c+d x))}{8 d}-\frac{i \tan ^{-1}\left (\frac{2 \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )}{\cosh \left (\frac{1}{2} (c+d x)\right )-2 \sinh \left (\frac{1}{2} (c+d x)\right )}\right )}{4 d}+\frac{i \tan ^{-1}\left (\frac{2 \sinh \left (\frac{1}{2} (c+d x)\right )+\cosh \left (\frac{1}{2} (c+d x)\right )}{\sinh \left (\frac{1}{2} (c+d x)\right )+2 \cosh \left (\frac{1}{2} (c+d x)\right )}\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 44, normalized size = 1.2 \begin{align*}{\frac{1}{4\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) }-{\frac{1}{4\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.76108, size = 49, normalized size = 1.32 \begin{align*} \frac{\log \left (-\frac{6 \,{\left (-i \, e^{\left (-d x - c\right )} + 3\right )}}{6 i \, e^{\left (-d x - c\right )} - 2}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02601, size = 80, normalized size = 2.16 \begin{align*} \frac{\log \left (e^{\left (d x + c\right )} - \frac{1}{3} i\right ) - \log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.599206, size = 31, normalized size = 0.84 \begin{align*} \frac{- \frac{\log{\left (e^{d x} - 3 i e^{- c} \right )}}{4} + \frac{\log{\left (e^{d x} - \frac{i e^{- c}}{3} \right )}}{4}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24547, size = 42, normalized size = 1.14 \begin{align*} \frac{\log \left (3 \, e^{\left (d x + c\right )} - i\right )}{4 \, d} - \frac{\log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]