Optimal. Leaf size=45 \[ \frac{\sqrt [3]{c \sin ^3(a+b x)}}{b^2}-\frac{x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
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Rubi [A] time = 0.127389, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6720, 3296, 2637} \[ \frac{\sqrt [3]{c \sin ^3(a+b x)}}{b^2}-\frac{x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x \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 \sin (a+b x) \, dx\\ &=-\frac{x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}+\frac{\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \cos (a+b x) \, dx}{b}\\ &=\frac{\sqrt [3]{c \sin ^3(a+b x)}}{b^2}-\frac{x \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.132428, size = 30, normalized size = 0.67 \[ \frac{(1-b x \cot (a+b x)) \sqrt [3]{c \sin ^3(a+b x)}}{b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.072, size = 117, normalized size = 2.6 \begin{align*}{\frac{-{\frac{i}{2}} \left ( bx+i \right ){{\rm e}^{2\,i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ){b}^{2}}\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 ( bx-i \right ) }{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ){b}^{2}}\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 = 1.51733, size = 81, normalized size = 1.8 \begin{align*} \frac{{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} c^{\frac{1}{3}} + \frac{4 \, a 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^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69519, size = 135, normalized size = 3. \begin{align*} -\frac{{\left (b x \cos \left (b x + a\right ) - \sin \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^{2} \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 = 5.14652, size = 76, normalized size = 1.69 \begin{align*} \begin{cases} \frac{x^{2} \sqrt [3]{c \sin ^{3}{\left (a \right )}}}{2} & \text{for}\: b = 0 \\0 & \text{for}\: a = - b x \vee a = - b x + \pi \\- \frac{\sqrt [3]{c} x \sqrt [3]{\sin ^{3}{\left (a + b x \right )}} \cos{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} + \frac{\sqrt [3]{c} \sqrt [3]{\sin ^{3}{\left (a + b x \right )}}}{b^{2}} & \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\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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