Optimal. Leaf size=135 \[ \sqrt {2 \pi } \sqrt {b} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\sqrt {2 \pi } \sqrt {b} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x} \]
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Rubi [A] time = 0.16, antiderivative size = 135, 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, 3387, 3354, 3352, 3351} \[ \sqrt {2 \pi } \sqrt {b} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\sqrt {2 \pi } \sqrt {b} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3387
Rule 6720
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^2} \, dx &=\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x}+\left (2 b \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\\ &=-\frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x}+\left (2 b \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-\left (2 b \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\\ &=-\frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x}+\sqrt {b} \sqrt {2 \pi } \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 )}-\sqrt {b} \sqrt {2 \pi } \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 )}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 105, normalized size = 0.78 \[ \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\sqrt {2 \pi } \sqrt {b} x \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right )-\sqrt {2 \pi } \sqrt {b} x \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \csc \left (a+b x^2\right )-1\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 150, normalized size = 1.11 \[ -\frac {4^{\frac {1}{3}} {\left (4^{\frac {2}{3}} \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) - 4^{\frac {2}{3}} \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (b x^{2} + a\right ) \sin \relax (a) + 4^{\frac {2}{3}} \cos \left (b x^{2} + a\right )^{2} - 4^{\frac {2}{3}}\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {1}{3}}}{4 \, {\left (x \cos \left (b x^{2} + a\right )^{2} - x\right )}} \]
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 \[ \int \frac {\left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {1}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 232, normalized size = 1.72 \[ \frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}} \left (-\frac {{\mathrm e}^{2 i \left (b \,x^{2}+a \right )}}{x}+\frac {i b \sqrt {\pi }\, \erf \left (\sqrt {-i b}\, x \right ) {\mathrm e}^{i \left (b \,x^{2}+2 a \right )}}{\sqrt {-i b}}\right )}{2 \,{\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-2}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}}}{2 x \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b \,x^{2}} b \sqrt {\pi }\, \erf \left (\sqrt {i b}\, x \right )}{2 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right ) \sqrt {i b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.45, size = 76, normalized size = 0.56 \[ \frac {\sqrt {b x^{2}} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \cos \relax (a) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \sin \relax (a)\right )} c^{\frac {1}{3}}}{16 \, x} \]
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 \[ \int \frac {{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{1/3}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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