Optimal. Leaf size=121 \[ -\frac{26 i a^2 \text{EllipticF}\left (\frac{i x}{2},2\right ) \sqrt{a \cosh ^3(x)}}{77 \cosh ^{\frac{3}{2}}(x)}+\frac{2}{15} a^2 \sinh (x) \cosh ^5(x) \sqrt{a \cosh ^3(x)}+\frac{26}{165} a^2 \sinh (x) \cosh ^3(x) \sqrt{a \cosh ^3(x)}+\frac{78}{385} a^2 \sinh (x) \cosh (x) \sqrt{a \cosh ^3(x)}+\frac{26}{77} a^2 \tanh (x) \sqrt{a \cosh ^3(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0538256, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2641} \[ \frac{2}{15} a^2 \sinh (x) \cosh ^5(x) \sqrt{a \cosh ^3(x)}+\frac{26}{165} a^2 \sinh (x) \cosh ^3(x) \sqrt{a \cosh ^3(x)}+\frac{78}{385} a^2 \sinh (x) \cosh (x) \sqrt{a \cosh ^3(x)}+\frac{26}{77} a^2 \tanh (x) \sqrt{a \cosh ^3(x)}-\frac{26 i a^2 F\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh ^3(x)}}{77 \cosh ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3207
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \left (a \cosh ^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \cosh ^3(x)}\right ) \int \cosh ^{\frac{15}{2}}(x) \, dx}{\cosh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{15} a^2 \cosh ^5(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{\left (13 a^2 \sqrt{a \cosh ^3(x)}\right ) \int \cosh ^{\frac{11}{2}}(x) \, dx}{15 \cosh ^{\frac{3}{2}}(x)}\\ &=\frac{26}{165} a^2 \cosh ^3(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{2}{15} a^2 \cosh ^5(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{\left (39 a^2 \sqrt{a \cosh ^3(x)}\right ) \int \cosh ^{\frac{7}{2}}(x) \, dx}{55 \cosh ^{\frac{3}{2}}(x)}\\ &=\frac{78}{385} a^2 \cosh (x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{26}{165} a^2 \cosh ^3(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{2}{15} a^2 \cosh ^5(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{\left (39 a^2 \sqrt{a \cosh ^3(x)}\right ) \int \cosh ^{\frac{3}{2}}(x) \, dx}{77 \cosh ^{\frac{3}{2}}(x)}\\ &=\frac{78}{385} a^2 \cosh (x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{26}{165} a^2 \cosh ^3(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{2}{15} a^2 \cosh ^5(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{26}{77} a^2 \sqrt{a \cosh ^3(x)} \tanh (x)+\frac{\left (13 a^2 \sqrt{a \cosh ^3(x)}\right ) \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{77 \cosh ^{\frac{3}{2}}(x)}\\ &=-\frac{26 i a^2 \sqrt{a \cosh ^3(x)} F\left (\left .\frac{i x}{2}\right |2\right )}{77 \cosh ^{\frac{3}{2}}(x)}+\frac{78}{385} a^2 \cosh (x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{26}{165} a^2 \cosh ^3(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{2}{15} a^2 \cosh ^5(x) \sqrt{a \cosh ^3(x)} \sinh (x)+\frac{26}{77} a^2 \sqrt{a \cosh ^3(x)} \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.114683, size = 65, normalized size = 0.54 \[ \frac{a \left (a \cosh ^3(x)\right )^{3/2} \left ((15465 \sinh (x)+3657 \sinh (3 x)+749 \sinh (5 x)+77 \sinh (7 x)) \sqrt{\cosh (x)}-12480 i \text{EllipticF}\left (\frac{i x}{2},2\right )\right )}{36960 \cosh ^{\frac{9}{2}}(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( \cosh \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{5}{2}}}\, 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 \left (a \cosh \left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \cosh \left (x\right )^{3}} a^{2} \cosh \left (x\right )^{6}, x\right ) \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 \left (a \cosh \left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]