Optimal. Leaf size=31 \[ -\frac{\cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
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Rubi [A] time = 0.104194, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6715, 3207, 2638} \[ -\frac{\cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 3207
Rule 2638
Rubi steps
\begin{align*} \int x \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt [3]{c \sin ^3(a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,x^2\right )\\ &=-\frac{\cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0517917, size = 31, normalized size = 1. \[ -\frac{\cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.071, size = 119, normalized size = 3.8 \begin{align*}{\frac{-{\frac{i}{4}}{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) b}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }}}}-{\frac{{\frac{i}{4}}}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) b}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53615, size = 22, normalized size = 0.71 \begin{align*} \frac{c^{\frac{1}{3}} \cos \left (b x^{2} + a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60993, size = 120, normalized size = 3.87 \begin{align*} -\frac{\left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{1}{3}} \cos \left (b x^{2} + a\right )}{2 \, b \sin \left (b x^{2} + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.09722, size = 66, normalized size = 2.13 \begin{align*} \begin{cases} 0 & \text{for}\: a = - b x^{2} \vee a = - b x^{2} + \pi \\\frac{x^{2} \sqrt [3]{c \sin ^{3}{\left (a \right )}}}{2} & \text{for}\: b = 0 \\- \frac{\sqrt [3]{c} \sqrt [3]{\sin ^{3}{\left (a + b x^{2} \right )}} \cos{\left (a + b x^{2} \right )}}{2 b \sin{\left (a + b x^{2} \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^{2} + 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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