Optimal. Leaf size=65 \[ \frac{1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b} \]
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Rubi [A] time = 0.0997761, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6715, 3207, 2635, 8} \[ \frac{1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 3207
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (c \sin ^3(a+b x)\right )^{2/3} \, dx,x,x^2\right )\\ &=\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 \sin ^2(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\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 1 \, dx,x,x^2\right )\\ &=-\frac{\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} x^2 \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 = 0.118257, size = 55, normalized size = 0.85 \[ \frac{\left (2 \left (a+b x^2\right )-\sin \left (2 \left (a+b x^2\right )\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.075, size = 182, normalized size = 2.8 \begin{align*} -{\frac{{x}^{2}{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{4\, \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}{16}}{{\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} \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}{16}}}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}b} \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.
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Maxima [A] time = 1.54614, size = 38, normalized size = 0.58 \begin{align*} -\frac{{\left (2 \, b x^{2} - \sin \left (2 \, b x^{2} + 2 \, a\right )\right )} c^{\frac{2}{3}}}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58751, size = 162, normalized size = 2.49 \begin{align*} -\frac{{\left (b x^{2} - \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right )\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}}}{4 \,{\left (b \cos \left (b x^{2} + a\right )^{2} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (c \sin ^{3}{\left (a + b x^{2} \right )}\right )^{\frac{2}{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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