Optimal. Leaf size=188 \[ \frac{e^{2 i a} 4^{-\frac{1}{n}-1} x^2 \left (-i b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 4^{-\frac{1}{n}-1} x^2 \left (i b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
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Rubi [A] time = 0.240431, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6720, 3425, 3424, 2218} \[ \frac{e^{2 i a} 4^{-\frac{1}{n}-1} x^2 \left (-i b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 4^{-\frac{1}{n}-1} x^2 \left (i b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3425
Rule 3424
Rule 2218
Rubi steps
\begin{align*} \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int x \sin ^2\left (a+b x^n\right ) \, dx\\ &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \left (\frac{x}{2}-\frac{1}{2} x \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac{1}{2} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int x \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac{1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac{1}{4} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int e^{-2 i a-2 i b x^n} x \, dx-\frac{1}{4} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int e^{2 i a+2 i b x^n} x \, dx\\ &=\frac{1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac{4^{-1-\frac{1}{n}} e^{2 i a} x^2 \left (-i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{2}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{4^{-1-\frac{1}{n}} e^{-2 i a} x^2 \left (i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{2}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}\\ \end{align*}
Mathematica [A] time = 0.548596, size = 160, normalized size = 0.85 \[ \frac{e^{-2 i a} 4^{-\frac{n+1}{n}} x^2 \left (b^2 x^{2 n}\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \left (e^{4 i a} \left (i b x^n\right )^{2/n} \text{Gamma}\left (\frac{2}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{2/n} \text{Gamma}\left (\frac{2}{n},2 i b x^n\right )+e^{2 i a} 4^{\frac{1}{n}} n \left (b^2 x^{2 n}\right )^{2/n}\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.23, size = 0, normalized size = 0. \begin{align*} \int x \left ( c \left ( \sin \left ( a+b{x}^{n} \right ) \right ) ^{3} \right ) ^{{\frac{2}{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*} -\frac{1}{8} \,{\left (x^{2} - 2 \, \int x \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}\right )} c^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac{2}{3}} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{n} + a\right )^{3}\right )^{\frac{2}{3}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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