Optimal. Leaf size=153 \[ \frac {1}{4} i e^{i a} x^{1+m} \left (-i b x^2\right )^{\frac {1}{2} (-1-m)} \csc \left (a+b x^2\right ) \Gamma \left (\frac {1+m}{2},-i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\frac {1}{4} i e^{-i a} x^{1+m} \left (i b x^2\right )^{\frac {1}{2} (-1-m)} \csc \left (a+b x^2\right ) \Gamma \left (\frac {1+m}{2},i b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \]
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Rubi [A]
time = 0.20, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6852, 3470,
2250} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3470
Rule 6852
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*}
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Mathematica [A]
time = 0.20, size = 138, normalized size = 0.90 \begin {gather*} \frac {1}{4} i x^{1+m} \left (b^2 x^4\right )^{\frac {1}{2} (-1-m)} \csc \left (a+b x^2\right ) \left (-\left (-i b x^2\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1+m}{2},i b x^2\right ) (\cos (a)-i \sin (a))+\left (i b x^2\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1+m}{2},-i b x^2\right ) (\cos (a)+i \sin (a))\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int x^{m} \left (c \left (\sin ^{3}\left (b \,x^{2}+a \right )\right )\right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 98, normalized size = 0.64 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \int x^{m} \sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^m\,{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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