Optimal. Leaf size=139 \[ \frac{x \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b^2}+\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^3}-\frac{x \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}-\frac{x^2 \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac{1}{6} x^3 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \]
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Rubi [A] time = 0.163858, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6720, 3311, 30, 2635, 8} \[ \frac{x \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b^2}+\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^3}-\frac{x \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}-\frac{x^2 \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac{1}{6} x^3 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3311
Rule 30
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x^2 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx &=\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int x^2 \sin ^2(a+b x) \, dx\\ &=\frac{x \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b^2}-\frac{x^2 \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac{1}{2} \left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int x^2 \, dx-\frac{\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \sin ^2(a+b x) \, dx}{2 b^2}\\ &=\frac{x \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b^2}+\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^3}-\frac{x^2 \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac{1}{6} x^3 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac{\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int 1 \, dx}{4 b^2}\\ &=\frac{x \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b^2}+\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^3}-\frac{x^2 \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}-\frac{x \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}+\frac{1}{6} x^3 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\\ \end{align*}
Mathematica [A] time = 0.281151, size = 69, normalized size = 0.5 \[ \frac{\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (\left (3-6 b^2 x^2\right ) \sin (2 (a+b x))-6 b x \cos (2 (a+b x))+4 b^3 x^3\right )}{24 b^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.078, size = 190, normalized size = 1.4 \begin{align*} -{\frac{{x}^{3}{{\rm e}^{2\,i \left ( bx+a \right ) }}}{6\, \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}}}-{\frac{{\frac{i}{16}} \left ( 2\,{x}^{2}{b}^{2}+2\,ibx-1 \right ){{\rm e}^{4\,i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}{b}^{3}} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}}}+{\frac{{\frac{i}{16}} \left ( 2\,{x}^{2}{b}^{2}-2\,ibx-1 \right ) }{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}{b}^{3}} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+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 = 1.58114, size = 296, normalized size = 2.13 \begin{align*} \frac{48 \,{\left (c^{\frac{2}{3}} \arctan \left (\frac{\sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1}\right ) - \frac{\frac{c^{\frac{2}{3}} \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} - \frac{c^{\frac{2}{3}} \sin \left (b x + a\right )^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}}{\frac{2 \, \sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{\sin \left (b x + a\right )^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1}\right )} a^{2} + 6 \,{\left (2 \,{\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right )\right )} a c^{\frac{2}{3}} -{\left (4 \,{\left (b x + a\right )}^{3} - 6 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \,{\left (2 \,{\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c^{\frac{2}{3}}}{48 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71005, size = 227, normalized size = 1.63 \begin{align*} -\frac{{\left (2 \, b^{3} x^{3} - 6 \, b x \cos \left (b x + a\right )^{2} - 3 \,{\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 \, b x\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{2}{3}}}{12 \,{\left (b^{3} \cos \left (b x + a\right )^{2} - b^{3}\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^{2} \left (c \sin ^{3}{\left (a + b x \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 + a\right )^{3}\right )^{\frac{2}{3}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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