Optimal. Leaf size=83 \[ \frac{2}{9} a \sinh ^2(x) \cosh (x) \sqrt{a \sinh ^3(x)}-\frac{14}{45} a \cosh (x) \sqrt{a \sinh ^3(x)}+\frac{14 i a \text{csch}(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \sinh ^3(x)}}{15 \sqrt{i \sinh (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0436829, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3207, 2635, 2640, 2639} \[ \frac{2}{9} a \sinh ^2(x) \cosh (x) \sqrt{a \sinh ^3(x)}-\frac{14}{45} a \cosh (x) \sqrt{a \sinh ^3(x)}+\frac{14 i a \text{csch}(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \sinh ^3(x)}}{15 \sqrt{i \sinh (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3207
Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \left (a \sinh ^3(x)\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a \sinh ^3(x)}\right ) \int \sinh ^{\frac{9}{2}}(x) \, dx}{\sinh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{9} a \cosh (x) \sinh ^2(x) \sqrt{a \sinh ^3(x)}-\frac{\left (7 a \sqrt{a \sinh ^3(x)}\right ) \int \sinh ^{\frac{5}{2}}(x) \, dx}{9 \sinh ^{\frac{3}{2}}(x)}\\ &=-\frac{14}{45} a \cosh (x) \sqrt{a \sinh ^3(x)}+\frac{2}{9} a \cosh (x) \sinh ^2(x) \sqrt{a \sinh ^3(x)}+\frac{\left (7 a \sqrt{a \sinh ^3(x)}\right ) \int \sqrt{\sinh (x)} \, dx}{15 \sinh ^{\frac{3}{2}}(x)}\\ &=-\frac{14}{45} a \cosh (x) \sqrt{a \sinh ^3(x)}+\frac{2}{9} a \cosh (x) \sinh ^2(x) \sqrt{a \sinh ^3(x)}+\frac{\left (7 a \text{csch}(x) \sqrt{a \sinh ^3(x)}\right ) \int \sqrt{i \sinh (x)} \, dx}{15 \sqrt{i \sinh (x)}}\\ &=-\frac{14}{45} a \cosh (x) \sqrt{a \sinh ^3(x)}+\frac{14 i a \text{csch}(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \sinh ^3(x)}}{15 \sqrt{i \sinh (x)}}+\frac{2}{9} a \cosh (x) \sinh ^2(x) \sqrt{a \sinh ^3(x)}\\ \end{align*}
Mathematica [A] time = 0.0644298, size = 57, normalized size = 0.69 \[ \frac{1}{180} a \text{csch}(x) \sqrt{a \sinh ^3(x)} \left (-38 \sinh (2 x)+5 \sinh (4 x)+168 \sqrt{i \sinh (x)} \text{csch}(x) E\left (\left .\frac{1}{4} (\pi -2 i x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( \sinh \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{3}{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 \sinh \left (x\right )^{3}\right )^{\frac{3}{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 \sinh \left (x\right )^{3}} a \sinh \left (x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sinh ^{3}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \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 \sinh \left (x\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]