Optimal. Leaf size=47 \[ \frac{4}{15 \cosh ^{\frac{3}{2}}(x)}-\frac{12 \sqrt{\cosh (x)}}{5}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{6 x \sinh (x)}{5 \sqrt{\cosh (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0708335, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3315} \[ \frac{4}{15 \cosh ^{\frac{3}{2}}(x)}-\frac{12 \sqrt{\cosh (x)}}{5}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{6 x \sinh (x)}{5 \sqrt{\cosh (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\cosh ^{\frac{7}{2}}(x)}+\frac{3}{5} x \sqrt{\cosh (x)}\right ) \, dx &=\frac{3}{5} \int x \sqrt{\cosh (x)} \, dx+\int \frac{x}{\cosh ^{\frac{7}{2}}(x)} \, dx\\ &=\frac{4}{15 \cosh ^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{3}{5} \int \frac{x}{\cosh ^{\frac{3}{2}}(x)} \, dx+\frac{3}{5} \int x \sqrt{\cosh (x)} \, dx\\ &=\frac{4}{15 \cosh ^{\frac{3}{2}}(x)}-\frac{12 \sqrt{\cosh (x)}}{5}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{6 x \sinh (x)}{5 \sqrt{\cosh (x)}}\\ \end{align*}
Mathematica [A] time = 0.583576, size = 64, normalized size = 1.36 \[ \frac{1}{5} \sqrt{\cosh (x)} \left (6 x \tanh (x)+\left (2 x \tanh (x)+\frac{4}{3}\right ) \text{sech}^2(x)-\frac{12 \sinh ^2(x)}{\sqrt{\cosh (x)-1} (\cosh (x)+1)^{3/2} \sqrt{\tanh ^2\left (\frac{x}{2}\right )}}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,x}{5}\sqrt{\cosh \left ( x \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3}{5} \, x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3}{5} \, x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]