Optimal. Leaf size=155 \[ \frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac {x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
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Rubi [A] time = 0.21, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6720, 3385, 3354, 3352, 3351} \[ \frac {\sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac {x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3385
Rule 6720
Rubi steps
\begin {align*} \int x^2 \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx &=\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int x^2 \sin \left (a+b x^2\right ) \, dx\\ &=-\frac {x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}+\frac {\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \cos \left (a+b x^2\right ) \, dx}{2 b}\\ &=-\frac {x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}+\frac {\left (\cos (a) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \cos \left (b x^2\right ) \, dx}{2 b}-\frac {\left (\csc \left (a+b x^2\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \sin \left (b x^2\right ) \, dx}{2 b}\\ &=-\frac {x \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \csc \left (a+b x^2\right ) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \csc \left (a+b x^2\right ) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 105, normalized size = 0.68 \[ -\frac {\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (-\sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\sqrt {2 \pi } \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+2 \sqrt {b} x \cos \left (a+b x^2\right )\right )}{4 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 156, normalized size = 1.01 \[ -\frac {4^{\frac {1}{3}} {\left (4^{\frac {2}{3}} \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) - 4^{\frac {2}{3}} \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) \sin \relax (a) - 2 \cdot 4^{\frac {2}{3}} b x \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {1}{3}}}{16 \, {\left (b^{2} \cos \left (b x^{2} + a\right )^{2} - b^{2}\right )}} \]
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 \[ \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {1}{3}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 240, normalized size = 1.55 \[ \frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}} \left (-\frac {i x \,{\mathrm e}^{2 i \left (b \,x^{2}+a \right )}}{2 b}+\frac {i \sqrt {\pi }\, \erf \left (\sqrt {-i b}\, x \right ) {\mathrm e}^{i \left (b \,x^{2}+2 a \right )}}{4 b \sqrt {-i b}}\right )}{2 \,{\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-2}-\frac {i x \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}}}{4 b \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b \,x^{2}} \sqrt {\pi }\, \erf \left (\sqrt {i b}\, x \right )}{8 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right ) b \sqrt {i b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.96, size = 73, normalized size = 0.47 \[ \frac {8 \, b^{2} c^{\frac {1}{3}} x \cos \left (b x^{2} + a\right ) + \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \relax (a) + \left (i + 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \relax (a) - \left (i - 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}} c^{\frac {1}{3}}}{32 \, b^{3}} \]
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 \[ \int x^2\,{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{1/3} \,d x \]
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 \[ \int x^{2} \sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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