Optimal. Leaf size=217 \[ \frac{e^{2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{x^{m+1} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (m+1)} \]
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Rubi [A] time = 0.364997, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 3425, 3424, 2218} \[ \frac{e^{2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{x^{m+1} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3425
Rule 3424
Rule 2218
Rubi steps
\begin{align*} \int x^m \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^m \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^m}{2}-\frac{1}{2} x^m \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}-\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^m \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}-\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^m \, 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^m \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}+\frac{2^{-\frac{1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac{1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{1+m}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{2^{-\frac{1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac{1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{1+m}{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 = 0.831941, size = 194, normalized size = 0.89 \[ \frac{e^{-2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \left (e^{4 i a} (m+1) \left (i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-2 i b x^n\right )+(m+1) \left (-i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},2 i b x^n\right )+e^{2 i a} n 2^{\frac{m+n+1}{n}} \left (b^2 x^{2 n}\right )^{\frac{m+1}{n}}\right )}{(m+1) n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.211, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \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{{\left (x x^{m} -{\left (m + 1\right )} \int x^{m} \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}\right )} c^{\frac{2}{3}}}{4 \,{\left (m + 1\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^{m}, 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^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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