Optimal. Leaf size=153 \[ \frac{1}{4} i e^{i a} x^{m+1} \left (-i b x^2\right )^{\frac{1}{2} (-m-1)} \csc \left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},-i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\frac{1}{4} i e^{-i a} x^{m+1} \left (i b x^2\right )^{\frac{1}{2} (-m-1)} \csc \left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \]
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Rubi [A] time = 0.301327, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6720, 3389, 2218} \[ \frac{1}{4} i e^{i a} x^{m+1} \left (-i b x^2\right )^{\frac{1}{2} (-m-1)} \csc \left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},-i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\frac{1}{4} i e^{-i a} x^{m+1} \left (i b x^2\right )^{\frac{1}{2} (-m-1)} \csc \left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3389
Rule 2218
Rubi steps
\begin{align*} \int x^m \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^m \sin \left (a+b x^2\right ) \, dx\\ &=\frac{1}{2} \left (i \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int e^{-i a-i b x^2} x^m \, dx-\frac{1}{2} \left (i \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int e^{i a+i b x^2} x^m \, dx\\ &=\frac{1}{4} i e^{i a} x^{1+m} \left (-i b x^2\right )^{\frac{1}{2} (-1-m)} \csc \left (a+b x^2\right ) \Gamma \left (\frac{1+m}{2},-i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\frac{1}{4} i e^{-i a} x^{1+m} \left (i b x^2\right )^{\frac{1}{2} (-1-m)} \csc \left (a+b x^2\right ) \Gamma \left (\frac{1+m}{2},i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.300039, size = 138, normalized size = 0.9 \[ \frac{1}{4} i x^{m+1} \left (b^2 x^4\right )^{\frac{1}{2} (-m-1)} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left ((\cos (a)+i \sin (a)) \left (i b x^2\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{m+1}{2},-i b x^2\right )-(\cos (a)-i \sin (a)) \left (-i b x^2\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{m+1}{2},i b x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt [3]{c \left ( \sin \left ( b{x}^{2}+a \right ) \right ) ^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac{1}{3}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80492, size = 270, normalized size = 1.76 \begin{align*} -\frac{{\left (e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, i \, b x^{2}\right ) + e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -i \, b x^{2}\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}}}{4 \, b \sin \left (b x^{2} + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac{1}{3}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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