Optimal. Leaf size=117 \[ \frac {\sqrt {\frac {\pi }{2}} \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}} \]
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Rubi [A] time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6720, 3353, 3352, 3351} \[ \frac {\sqrt {\frac {\pi }{2}} \sin (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 6720
Rubi steps
\begin {align*} \int \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 \sin \left (a+b x^2\right ) \, dx\\ &=\left (\cos (a) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \sin \left (b x^2\right ) \, dx+\left (\csc \left (a+b x^2\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \cos \left (b x^2\right ) \, dx\\ &=\frac {\sqrt {\frac {\pi }{2}} \cos (a) \csc \left (a+b x^2\right ) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \csc \left (a+b x^2\right ) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 80, normalized size = 0.68 \[ \frac {\sqrt {\frac {\pi }{2}} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 128, normalized size = 1.09 \[ -\frac {4^{\frac {1}{3}} {\left (4^{\frac {2}{3}} \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) + 4^{\frac {2}{3}} \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) \sin \relax (a)\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {1}{3}}}{8 \, {\left (b \cos \left (b x^{2} + a\right )^{2} - b\right )}} \]
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 \[ \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 157, normalized size = 1.34 \[ \frac {\erf \left (\sqrt {-i b}\, x \right ) \sqrt {\pi }\, \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i \left (b \,x^{2}+2 a \right )}}{4 \sqrt {-i b}\, \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )}-\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b \,x^{2}} \sqrt {\pi }\, \erf \left (\sqrt {i b}\, x \right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right ) \sqrt {i b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.93, size = 51, normalized size = 0.44 \[ \frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \relax (a) + \left (i - 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (\left (i - 1\right ) \, \cos \relax (a) - \left (i + 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} c^{\frac {1}{3}}}{16 \, \sqrt {b}} \]
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 \[ \int {\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{1/3} \,d x \]
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 \[ \int \sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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