Optimal. Leaf size=195 \[ \frac {\sqrt {\pi } \sin (2 a) C\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac {\sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}-\frac {x \sin \left (2 a+2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b}+\frac {1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]
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Rubi [A] time = 0.23, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6720, 3403, 3386, 3353, 3352, 3351} \[ \frac {\sqrt {\pi } \sin (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac {\sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac {1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac {x \sin \left (2 a+2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3386
Rule 3403
Rule 6720
Rubi steps
\begin {align*} \int x^2 \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^2 \sin ^2\left (a+b x^2\right ) \, 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 \left (\frac {x^2}{2}-\frac {1}{2} x^2 \cos \left (2 a+2 b x^2\right )\right ) \, dx\\ &=\frac {1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\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 ) \int x^2 \cos \left (2 a+2 b x^2\right ) \, dx\\ &=\frac {1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac {x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \sin \left (2 a+2 b x^2\right )}{8 b}+\frac {\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \sin \left (2 a+2 b x^2\right ) \, dx}{8 b}\\ &=\frac {1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac {x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \sin \left (2 a+2 b x^2\right )}{8 b}+\frac {\left (\cos (2 a) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \sin \left (2 b x^2\right ) \, dx}{8 b}+\frac {\left (\csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \cos \left (2 b x^2\right ) \, dx}{8 b}\\ &=\frac {1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {\sqrt {\pi } \cos (2 a) \csc ^2\left (a+b x^2\right ) S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac {\sqrt {\pi } \csc ^2\left (a+b x^2\right ) C\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}-\frac {x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \sin \left (2 a+2 b x^2\right )}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 113, normalized size = 0.58 \[ \frac {\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (3 \sqrt {\pi } \sin (2 a) C\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )+3 \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )+2 \sqrt {b} x \left (4 b x^2-3 \sin \left (2 \left (a+b x^2\right )\right )\right )\right )}{48 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 146, normalized size = 0.75 \[ -\frac {4^{\frac {2}{3}} {\left (8 \cdot 4^{\frac {1}{3}} b^{2} x^{3} - 12 \cdot 4^{\frac {1}{3}} b x \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right ) + 3 \cdot 4^{\frac {1}{3}} \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x \sqrt {\frac {b}{\pi }}\right ) + 3 \cdot 4^{\frac {1}{3}} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {2}{3}}}{192 \, {\left (b^{2} \cos \left (b x^{2} + a\right )^{2} - b^{2}\right )}} \]
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 \[ \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {2}{3}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 309, normalized size = 1.58 \[ \frac {i x \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}}}{16 b \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{2}} \sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, \sqrt {i b}\, x \right )}{64 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2} b \sqrt {i b}}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} \left (-\frac {i x \,{\mathrm e}^{4 i \left (b \,x^{2}+a \right )}}{4 b}+\frac {i \sqrt {\pi }\, \erf \left (\sqrt {-2 i b}\, x \right ) {\mathrm e}^{2 i \left (b \,x^{2}+2 a \right )}}{8 b \sqrt {-2 i b}}\right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {x^{3} \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{2}+a \right )}}{6 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.66, size = 98, normalized size = 0.50 \[ \frac {4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (3 i + 3\right ) \, \cos \left (2 \, a\right ) + \left (3 i - 3\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x\right ) + {\left (\left (3 i - 3\right ) \, \cos \left (2 \, a\right ) - \left (3 i + 3\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x\right )\right )} b^{\frac {3}{2}} c^{\frac {2}{3}} - 16 \, {\left (4 \, b^{3} x^{3} - 3 \, b^{2} x \sin \left (2 \, b x^{2} + 2 \, a\right )\right )} c^{\frac {2}{3}}}{768 \, b^{3}} \]
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 \[ \int x^2\,{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{2/3} \,d x \]
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 \[ \int x^{2} \left (c \sin ^{3}{\left (a + b x^{2} \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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