Optimal. Leaf size=74 \[ \frac{2 x \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac{2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac{x^2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
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Rubi [A] time = 0.182098, antiderivative size = 74, 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, 3296, 2638} \[ \frac{2 x \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac{2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac{x^2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^2 \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^2 \sin (a+b x) \, dx\\ &=-\frac{x^2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}+\frac{\left (2 \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int x \cos (a+b x) \, dx}{b}\\ &=\frac{2 x \sqrt [3]{c \sin ^3(a+b x)}}{b^2}-\frac{x^2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}-\frac{\left (2 \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \sin (a+b x) \, dx}{b^2}\\ &=\frac{2 x \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac{2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac{x^2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.223712, size = 40, normalized size = 0.54 \[ \frac{\left (\left (2-b^2 x^2\right ) \cot (a+b x)+2 b x\right ) \sqrt [3]{c \sin ^3(a+b x)}}{b^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.075, size = 133, normalized size = 1.8 \begin{align*}{\frac{-{\frac{i}{2}} \left ({x}^{2}{b}^{2}+2\,ibx-2 \right ){{\rm e}^{2\,i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ){b}^{3}}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}}}-{\frac{{\frac{i}{2}} \left ({x}^{2}{b}^{2}-2\,ibx-2 \right ) }{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ){b}^{3}}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54939, size = 134, normalized size = 1.81 \begin{align*} -\frac{2 \,{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a c^{\frac{1}{3}} -{\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \,{\left (b x + a\right )} \sin \left (b x + a\right )\right )} c^{\frac{1}{3}} + \frac{4 \, a^{2} c^{\frac{1}{3}}}{\frac{\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75473, size = 155, normalized size = 2.09 \begin{align*} \frac{{\left (2 \, b x \sin \left (b x + a\right ) -{\left (b^{2} x^{2} - 2\right )} \cos \left (b x + a\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{1}{3}}}{b^{3} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.8304, size = 117, normalized size = 1.58 \begin{align*} \begin{cases} \frac{x^{3} \sqrt [3]{c \sin ^{3}{\left (a \right )}}}{3} & \text{for}\: b = 0 \\0 & \text{for}\: a = - b x \vee a = - b x + \pi \\- \frac{\sqrt [3]{c} x^{2} \sqrt [3]{\sin ^{3}{\left (a + b x \right )}} \cos{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} + \frac{2 \sqrt [3]{c} x \sqrt [3]{\sin ^{3}{\left (a + b x \right )}}}{b^{2}} + \frac{2 \sqrt [3]{c} \sqrt [3]{\sin ^{3}{\left (a + b x \right )}} \cos{\left (a + b x \right )}}{b^{3} \sin{\left (a + b x \right )}} & \text{otherwise} \end{cases} \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^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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