Optimal. Leaf size=14 \[ -\frac{1}{3} (-\sinh (x)+i)^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0335453, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 32} \[ -\frac{1}{3} (-\sinh (x)+i)^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\cosh ^5(x)}{(i+\sinh (x))^2} \, dx &=\operatorname{Subst}\left (\int (i-x)^2 \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{3} (i-\sinh (x))^3\\ \end{align*}
Mathematica [A] time = 0.0193989, size = 18, normalized size = 1.29 \[ \frac{1}{6} \sinh (x) (-6 i \sinh (x)+\cosh (2 x)-7) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.047, size = 70, normalized size = 5. \begin{align*}{{\frac{1}{2}}-i \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{1+i \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{1-i \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{{\frac{1}{2}}+i \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.2142, size = 53, normalized size = 3.79 \begin{align*} -\frac{1}{96} \,{\left (24 i \, e^{\left (-x\right )} + 60 \, e^{\left (-2 \, x\right )} - 4\right )} e^{\left (3 \, x\right )} + \frac{5}{8} \, e^{\left (-x\right )} - \frac{1}{4} i \, e^{\left (-2 \, x\right )} - \frac{1}{24} \, e^{\left (-3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.97653, size = 107, normalized size = 7.64 \begin{align*} \frac{1}{24} \,{\left (e^{\left (6 \, x\right )} - 6 i \, e^{\left (5 \, x\right )} - 15 \, e^{\left (4 \, x\right )} + 15 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.315745, size = 44, normalized size = 3.14 \begin{align*} \frac{e^{3 x}}{24} - \frac{i e^{2 x}}{4} - \frac{5 e^{x}}{8} + \frac{5 e^{- x}}{8} - \frac{i e^{- 2 x}}{4} - \frac{e^{- 3 x}}{24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.21791, size = 47, normalized size = 3.36 \begin{align*} \frac{1}{24} \,{\left (15 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} + \frac{1}{24} \, e^{\left (3 \, x\right )} - \frac{1}{4} i \, e^{\left (2 \, x\right )} - \frac{5}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]