Optimal. Leaf size=15 \[ \frac{\sinh ^2(x)}{2}-i \sinh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0342303, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2667} \[ \frac{\sinh ^2(x)}{2}-i \sinh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{i+\sinh (x)} \, dx &=-\operatorname{Subst}(\int (i-x) \, dx,x,\sinh (x))\\ &=-i \sinh (x)+\frac{\sinh ^2(x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0102805, size = 12, normalized size = 0.8 \[ \frac{1}{2} \sinh (x) (\sinh (x)-2 i) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 13, normalized size = 0.9 \begin{align*} -i\sinh \left ( x \right ) +{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.06961, size = 36, normalized size = 2.4 \begin{align*} \frac{1}{8} \,{\left (-4 i \, e^{\left (-x\right )} + 1\right )} e^{\left (2 \, x\right )} + \frac{1}{2} i \, e^{\left (-x\right )} + \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7818, size = 70, normalized size = 4.67 \begin{align*} \frac{1}{8} \,{\left (e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} + 4 i \, e^{x} + 1\right )} e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.207659, size = 27, normalized size = 1.8 \begin{align*} \frac{e^{2 x}}{8} - \frac{i e^{x}}{2} + \frac{i e^{- x}}{2} + \frac{e^{- 2 x}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23027, size = 31, normalized size = 2.07 \begin{align*} -\frac{1}{8} \,{\left (-4 i \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{2} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]