Optimal. Leaf size=115 \[ -\frac{e^{i a} x^m (-i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,-i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b} \]
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Rubi [A] time = 0.287303, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6720, 3308, 2181} \[ -\frac{e^{i a} x^m (-i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,-i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int x^m \sin (a+b x) \, dx\\ &=\frac{1}{2} \left (i \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int e^{-i (a+b x)} x^m \, dx-\frac{1}{2} \left (i \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int e^{i (a+b x)} x^m \, dx\\ &=-\frac{e^{i a} x^m (-i b x)^{-m} \csc (a+b x) \Gamma (1+m,-i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \csc (a+b x) \Gamma (1+m,i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.125477, size = 94, normalized size = 0.82 \[ -\frac{e^{-i a} x^m \left (b^2 x^2\right )^{-m} \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (e^{2 i a} (i b x)^m \text{Gamma}(m+1,-i b x)+(-i b x)^m \text{Gamma}(m+1,i b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.154, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt [3]{c \left ( \sin \left ( bx+a \right ) \right ) ^{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*} \int \left (c \sin \left (b x + a\right )^{3}\right )^{\frac{1}{3}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74468, size = 213, normalized size = 1.85 \begin{align*} -\frac{{\left (e^{\left (-m \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 1, i \, b x\right ) + e^{\left (-m \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m + 1, -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{1}{3}}}{2 \, b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}\, dx \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{1}{3}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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