Optimal. Leaf size=169 \[ \frac{i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \csc ^2(a+b x) \text{Gamma}(m+1,-2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b}-\frac{i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \csc ^2(a+b x) \text{Gamma}(m+1,2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b}+\frac{x^{m+1} \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 (m+1)} \]
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Rubi [A] time = 0.301069, antiderivative size = 169, 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, 3312, 3307, 2181} \[ \frac{i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \csc ^2(a+b x) \text{Gamma}(m+1,-2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b}-\frac{i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \csc ^2(a+b x) \text{Gamma}(m+1,2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b}+\frac{x^{m+1} \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^m \left (c \sin ^3(a+b x)\right )^{2/3} \, dx &=\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int x^m \sin ^2(a+b x) \, dx\\ &=\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \left (\frac{x^m}{2}-\frac{1}{2} x^m \cos (2 a+2 b x)\right ) \, dx\\ &=\frac{x^{1+m} \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 (1+m)}-\frac{1}{2} \left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int x^m \cos (2 a+2 b x) \, dx\\ &=\frac{x^{1+m} \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 (1+m)}-\frac{1}{4} \left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int e^{-i (2 a+2 b x)} x^m \, dx-\frac{1}{4} \left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int e^{i (2 a+2 b x)} x^m \, dx\\ &=\frac{x^{1+m} \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 (1+m)}+\frac{i 2^{-3-m} e^{2 i a} x^m (-i b x)^{-m} \csc ^2(a+b x) \Gamma (1+m,-2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b}-\frac{i 2^{-3-m} e^{-2 i a} x^m (i b x)^{-m} \csc ^2(a+b x) \Gamma (1+m,2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b}\\ \end{align*}
Mathematica [A] time = 0.536991, size = 142, normalized size = 0.84 \[ \frac{2^{-m-3} x^m \left (b^2 x^2\right )^{-m} \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (-i (m+1) (\cos (a)-i \sin (a))^2 (-i b x)^m \text{Gamma}(m+1,2 i b x)+i (m+1) (\cos (a)+i \sin (a))^2 (i b x)^m \text{Gamma}(m+1,-2 i b x)+b 2^{m+2} x \left (b^2 x^2\right )^m\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.134, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( c \left ( \sin \left ( bx+a \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 ({\left (m + 1\right )} \int x^{m} \cos \left (2 \, b x + 2 \, a\right )\,{d x} - e^{\left (m \log \left (x\right ) + \log \left (x\right )\right )}\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 [A] time = 1.86369, size = 301, normalized size = 1.78 \begin{align*} -\frac{{\left (4 \, b x x^{m} -{\left (i \, m + i\right )} e^{\left (-m \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 1, 2 i \, b x\right ) -{\left (-i \, m - i\right )} e^{\left (-m \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 1, -2 i \, b x\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{2}{3}}}{8 \,{\left ({\left (b m + b\right )} \cos \left (b x + a\right )^{2} - b m - b\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 + 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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