Optimal. Leaf size=91 \[ \frac{\left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b^2}-\frac{x^2 \cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac{1}{8} x^4 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.183493, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 3379, 3310, 30} \[ \frac{\left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b^2}-\frac{x^2 \cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac{1}{8} x^4 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6720
Rule 3379
Rule 3310
Rule 30
Rubi steps
\begin{align*} \int x^3 \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int x^3 \sin ^2\left (a+b x^2\right ) \, dx\\ &=\frac{1}{2} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int x \sin ^2(a+b x) \, dx,x,x^2\right )\\ &=\frac{\left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b^2}-\frac{x^2 \cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac{1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int x \, dx,x,x^2\right )\\ &=\frac{\left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b^2}-\frac{x^2 \cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac{1}{8} x^4 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\\ \end{align*}
Mathematica [A] time = 0.233949, size = 67, normalized size = 0.74 \[ -\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (2 b x^2 \left (\sin \left (2 \left (a+b x^2\right )\right )-b x^2\right )+\cos \left (2 \left (a+b x^2\right )\right )\right )}{16 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.086, size = 200, normalized size = 2.2 \begin{align*} -{\frac{{x}^{4}{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{8\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}}-{\frac{{\frac{i}{32}} \left ( 2\,b{x}^{2}+i \right ){{\rm e}^{4\,i \left ( b{x}^{2}+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}{b}^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}}+{\frac{{\frac{i}{32}} \left ( 2\,b{x}^{2}-i \right ) }{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}{b}^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.55119, size = 63, normalized size = 0.69 \begin{align*} -\frac{{\left (2 \, b^{2} x^{4} - 2 \, b x^{2} \sin \left (2 \, b x^{2} + 2 \, a\right ) - \cos \left (2 \, b x^{2} + 2 \, a\right )\right )} c^{\frac{2}{3}}}{32 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59906, size = 219, normalized size = 2.41 \begin{align*} -\frac{{\left (2 \, b^{2} x^{4} - 4 \, b x^{2} \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right ) - 2 \, \cos \left (b x^{2} + a\right )^{2} + 1\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{2}{3}}}{16 \,{\left (b^{2} \cos \left (b x^{2} + a\right )^{2} - b^{2}\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 (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac{2}{3}} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]