Optimal. Leaf size=58 \[ \frac{\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^2}-\frac{x^2 \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
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Rubi [A] time = 0.181158, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 3379, 3296, 2637} \[ \frac{\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^2}-\frac{x^2 \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3379
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \, dx &=\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \int x^3 \sin \left (a+b x^2\right ) \, dx\\ &=\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 x \sin (a+b x) \, dx,x,x^2\right )\\ &=-\frac{x^2 \cot \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b}+\frac{\left (\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,x^2\right )}{2 b}\\ &=\frac{\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^2}-\frac{x^2 \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 = 0.0909529, size = 38, normalized size = 0.66 \[ -\frac{\left (b x^2 \cot \left (a+b x^2\right )-1\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.08, size = 135, normalized size = 2.3 \begin{align*}{\frac{-{\frac{i}{4}} \left ( b{x}^{2}+i \right ){{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ){b}^{2}}\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 ( b{x}^{2}-i \right ) }{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ){b}^{2}}\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 = 1.54007, size = 43, normalized size = 0.74 \begin{align*} \frac{{\left (b x^{2} \cos \left (b x^{2} + a\right ) - \sin \left (b x^{2} + a\right )\right )} c^{\frac{1}{3}}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60669, size = 157, normalized size = 2.71 \begin{align*} -\frac{{\left (b x^{2} \cos \left (b x^{2} + a\right ) - \sin \left (b x^{2} + a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{1}{3}}}{2 \, b^{2} \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 = 30.2861, size = 92, normalized size = 1.59 \begin{align*} \begin{cases} 0 & \text{for}\: a = - b x^{2} \vee a = - b x^{2} + \pi \\\frac{x^{4} \sqrt [3]{c \sin ^{3}{\left (a \right )}}}{4} & \text{for}\: b = 0 \\- \frac{\sqrt [3]{c} x^{2} \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 )}} + \frac{\sqrt [3]{c} \sqrt [3]{\sin ^{3}{\left (a + b x^{2} \right )}}}{2 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^{2} + a\right )^{3}\right )^{\frac{1}{3}} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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