Optimal. Leaf size=178 \[ \frac{e^{2 i a} 2^{-\frac{1}{n}-2} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{1}{n}-2} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
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Rubi [A] time = 0.0888038, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6720, 3367, 3366, 2208} \[ \frac{e^{2 i a} 2^{-\frac{1}{n}-2} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{1}{n}-2} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3367
Rule 3366
Rule 2208
Rubi steps
\begin{align*} \int \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 \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{1}{2}-\frac{1}{2} \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac{1}{2} x \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 \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac{1}{2} x \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} \, 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} \, dx\\ &=\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac{2^{-2-\frac{1}{n}} e^{2 i a} x \left (-i b x^n\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{2^{-2-\frac{1}{n}} e^{-2 i a} x \left (i b x^n\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{1}{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 = 0.266812, size = 149, normalized size = 0.84 \[ \frac{e^{-2 i a} 2^{-\frac{1}{n}-2} x \left (b^2 x^{2 n}\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \left (e^{4 i a} \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},2 i b x^n\right )+e^{2 i a} 2^{\frac{1}{n}+1} n \left (b^2 x^{2 n}\right )^{\frac{1}{n}}\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.173, size = 0, normalized size = 0. \begin{align*} \int \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}{4} \, c^{\frac{2}{3}}{\left (x - \int \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}\right )} \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\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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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