Optimal. Leaf size=139 \[ \frac {\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^3}+\frac {x \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b^2}-\frac {x \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}+\frac {1}{6} x^3 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac {x^2 \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b} \]
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Rubi [A] time = 0.16, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 8
Rule 30
Rule 2635
Rule 3311
Rule 6720
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*}
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Mathematica [A] time = 0.28, size = 69, normalized size = 0.50 \[ \frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (\left (3-6 b^2 x^2\right ) \sin (2 (a+b x))-6 b x \cos (2 (a+b x))+4 b^3 x^3\right )}{24 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 95, normalized size = 0.68 \[ -\frac {{\left (2 \, b^{3} x^{3} - 6 \, b x \cos \left (b x + a\right )^{2} - 3 \, {\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 \, b x\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{12 \, {\left (b^{3} \cos \left (b x + a\right )^{2} - b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin \left (b x + a\right )^{3}\right )^{\frac {2}{3}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 190, normalized size = 1.37 \[ -\frac {x^{3} \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{6 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {i \left (2 x^{2} b^{2}+2 i b x -1\right ) \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{4 i \left (b x +a \right )}}{16 b^{3} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} \left (2 x^{2} b^{2}-2 i b x -1\right )}{16 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 219, normalized size = 1.58 \[ \frac {48 \, {\left (c^{\frac {2}{3}} \arctan \left (\frac {\sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1}\right ) - \frac {\frac {c^{\frac {2}{3}} \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} - \frac {c^{\frac {2}{3}} \sin \left (b x + a\right )^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}}{\frac {2 \, \sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {\sin \left (b x + a\right )^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1}\right )} a^{2} + 6 \, {\left (2 \, {\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right )\right )} a c^{\frac {2}{3}} - {\left (4 \, {\left (b x + a\right )}^{3} - 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c^{\frac {2}{3}}}{48 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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