Optimal. Leaf size=145 \[ \frac{4}{3} a x^2 \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{8}{3} a x^2 \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{16}{9} a x \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{32}{3} a x \sqrt{a \cosh (x)+a}+\frac{224}{9} a \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{32}{27} a \sinh ^2\left (\frac{x}{2}\right ) \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.149125, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3319, 3311, 3296, 2637, 2633} \[ \frac{4}{3} a x^2 \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{8}{3} a x^2 \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{16}{9} a x \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{32}{3} a x \sqrt{a \cosh (x)+a}+\frac{224}{9} a \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{32}{27} a \sinh ^2\left (\frac{x}{2}\right ) \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3319
Rule 3311
Rule 3296
Rule 2637
Rule 2633
Rubi steps
\begin{align*} \int x^2 (a+a \cosh (x))^{3/2} \, dx &=\left (2 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x^2 \cosh ^3\left (\frac{x}{2}\right ) \, dx\\ &=-\frac{16}{9} a x \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x^2 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{1}{3} \left (4 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x^2 \cosh \left (\frac{x}{2}\right ) \, dx+\frac{1}{9} \left (16 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \cosh ^3\left (\frac{x}{2}\right ) \, dx\\ &=-\frac{16}{9} a x \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x^2 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{8}{3} a x^2 \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{1}{9} \left (32 i a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh \left (\frac{x}{2}\right )\right )-\frac{1}{3} \left (16 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x \sinh \left (\frac{x}{2}\right ) \, dx\\ &=-\frac{32}{3} a x \sqrt{a+a \cosh (x)}-\frac{16}{9} a x \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x^2 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{32}{9} a \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{8}{3} a x^2 \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{32}{27} a \sqrt{a+a \cosh (x)} \sinh ^2\left (\frac{x}{2}\right ) \tanh \left (\frac{x}{2}\right )+\frac{1}{3} \left (32 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \cosh \left (\frac{x}{2}\right ) \, dx\\ &=-\frac{32}{3} a x \sqrt{a+a \cosh (x)}-\frac{16}{9} a x \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x^2 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{224}{9} a \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{8}{3} a x^2 \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{32}{27} a \sqrt{a+a \cosh (x)} \sinh ^2\left (\frac{x}{2}\right ) \tanh \left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.21464, size = 54, normalized size = 0.37 \[ \frac{2}{27} a \sqrt{a (\cosh (x)+1)} \left (\left (45 x^2+328\right ) \tanh \left (\frac{x}{2}\right )+\cosh (x) \left (\left (9 x^2+8\right ) \tanh \left (\frac{x}{2}\right )-12 x\right )-156 x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+a\cosh \left ( x \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.68322, size = 184, normalized size = 1.27 \begin{align*} -\frac{1}{54} \,{\left (9 \, \sqrt{2} a^{\frac{3}{2}} x^{2} + 12 \, \sqrt{2} a^{\frac{3}{2}} x + 8 \, \sqrt{2} a^{\frac{3}{2}} -{\left (9 \, \sqrt{2} a^{\frac{3}{2}} x^{2} - 12 \, \sqrt{2} a^{\frac{3}{2}} x + 8 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{\left (3 \, x\right )} - 81 \,{\left (\sqrt{2} a^{\frac{3}{2}} x^{2} - 4 \, \sqrt{2} a^{\frac{3}{2}} x + 8 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{\left (2 \, x\right )} + 81 \,{\left (\sqrt{2} a^{\frac{3}{2}} x^{2} + 4 \, \sqrt{2} a^{\frac{3}{2}} x + 8 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{x}\right )} e^{\left (-\frac{3}{2} \, x\right )} \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 [A] time = 1.20157, size = 176, normalized size = 1.21 \begin{align*} \frac{1}{54} \, \sqrt{2}{\left (9 \, a^{\frac{3}{2}} x^{2} e^{\left (\frac{3}{2} \, x\right )} + 81 \, a^{\frac{3}{2}} x^{2} e^{\left (\frac{1}{2} \, x\right )} - 81 \, a^{\frac{3}{2}} x^{2} e^{\left (-\frac{1}{2} \, x\right )} - 9 \, a^{\frac{3}{2}} x^{2} e^{\left (-\frac{3}{2} \, x\right )} - 12 \, a^{\frac{3}{2}} x e^{\left (\frac{3}{2} \, x\right )} - 324 \, a^{\frac{3}{2}} x e^{\left (\frac{1}{2} \, x\right )} - 324 \, a^{\frac{3}{2}} x e^{\left (-\frac{1}{2} \, x\right )} - 12 \, a^{\frac{3}{2}} x e^{\left (-\frac{3}{2} \, x\right )} + 8 \, a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} + 648 \, a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - 648 \, a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, x\right )} - 8 \, a^{\frac{3}{2}} e^{\left (-\frac{3}{2} \, x\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]