Optimal. Leaf size=19 \[ \frac{\cosh ^5(x)}{5}-\frac{2 \cosh ^3(x)}{3}+\cosh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01108, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2633} \[ \frac{\cosh ^5(x)}{5}-\frac{2 \cosh ^3(x)}{3}+\cosh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2633
Rubi steps
\begin{align*} \int \sinh ^5(x) \, dx &=\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\frac{2 \cosh ^3(x)}{3}+\frac{\cosh ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0019877, size = 23, normalized size = 1.21 \[ \frac{5 \cosh (x)}{8}-\frac{5}{48} \cosh (3 x)+\frac{1}{80} \cosh (5 x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 18, normalized size = 1. \begin{align*} \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.93214, size = 47, normalized size = 2.47 \begin{align*} \frac{1}{160} \, e^{\left (5 \, x\right )} - \frac{5}{96} \, e^{\left (3 \, x\right )} + \frac{5}{16} \, e^{\left (-x\right )} - \frac{5}{96} \, e^{\left (-3 \, x\right )} + \frac{1}{160} \, e^{\left (-5 \, x\right )} + \frac{5}{16} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.10196, size = 155, normalized size = 8.16 \begin{align*} \frac{1}{80} \, \cosh \left (x\right )^{5} + \frac{1}{16} \, \cosh \left (x\right ) \sinh \left (x\right )^{4} - \frac{5}{48} \, \cosh \left (x\right )^{3} + \frac{1}{16} \,{\left (2 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac{5}{8} \, \cosh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.12174, size = 29, normalized size = 1.53 \begin{align*} \sinh ^{4}{\left (x \right )} \cosh{\left (x \right )} - \frac{4 \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} + \frac{8 \cosh ^{5}{\left (x \right )}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.10703, size = 50, normalized size = 2.63 \begin{align*} \frac{1}{480} \,{\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac{1}{160} \, e^{\left (5 \, x\right )} - \frac{5}{96} \, e^{\left (3 \, x\right )} + \frac{5}{16} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]