Optimal. Leaf size=60 \[ \frac{1}{6} x^3 \left (2 a^2+b^2\right )-\frac{2 a b \cos \left (c+d x^3\right )}{3 d}-\frac{b^2 \sin \left (c+d x^3\right ) \cos \left (c+d x^3\right )}{6 d} \]
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Rubi [A] time = 0.0569236, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3379, 2644} \[ \frac{1}{6} x^3 \left (2 a^2+b^2\right )-\frac{2 a b \cos \left (c+d x^3\right )}{3 d}-\frac{b^2 \sin \left (c+d x^3\right ) \cos \left (c+d x^3\right )}{6 d} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 2644
Rubi steps
\begin{align*} \int x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int (a+b \sin (c+d x))^2 \, dx,x,x^3\right )\\ &=\frac{1}{6} \left (2 a^2+b^2\right ) x^3-\frac{2 a b \cos \left (c+d x^3\right )}{3 d}-\frac{b^2 \cos \left (c+d x^3\right ) \sin \left (c+d x^3\right )}{6 d}\\ \end{align*}
Mathematica [A] time = 0.146364, size = 52, normalized size = 0.87 \[ -\frac{-2 \left (2 a^2+b^2\right ) \left (c+d x^3\right )+8 a b \cos \left (c+d x^3\right )+b^2 \sin \left (2 \left (c+d x^3\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 62, normalized size = 1. \begin{align*}{\frac{1}{3\,d} \left ({b}^{2} \left ( -{\frac{\cos \left ( d{x}^{3}+c \right ) \sin \left ( d{x}^{3}+c \right ) }{2}}+{\frac{d{x}^{3}}{2}}+{\frac{c}{2}} \right ) -2\,ab\cos \left ( d{x}^{3}+c \right ) +{a}^{2} \left ( d{x}^{3}+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98571, size = 70, normalized size = 1.17 \begin{align*} \frac{1}{3} \, a^{2} x^{3} + \frac{{\left (2 \, d x^{3} - \sin \left (2 \, d x^{3} + 2 \, c\right )\right )} b^{2}}{12 \, d} - \frac{2 \, a b \cos \left (d x^{3} + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66223, size = 119, normalized size = 1.98 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )} d x^{3} - b^{2} \cos \left (d x^{3} + c\right ) \sin \left (d x^{3} + c\right ) - 4 \, a b \cos \left (d x^{3} + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24779, size = 99, normalized size = 1.65 \begin{align*} \begin{cases} \frac{a^{2} x^{3}}{3} - \frac{2 a b \cos{\left (c + d x^{3} \right )}}{3 d} + \frac{b^{2} x^{3} \sin ^{2}{\left (c + d x^{3} \right )}}{6} + \frac{b^{2} x^{3} \cos ^{2}{\left (c + d x^{3} \right )}}{6} - \frac{b^{2} \sin{\left (c + d x^{3} \right )} \cos{\left (c + d x^{3} \right )}}{6 d} & \text{for}\: d \neq 0 \\\frac{x^{3} \left (a + b \sin{\left (c \right )}\right )^{2}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0961, size = 77, normalized size = 1.28 \begin{align*} \frac{4 \,{\left (d x^{3} + c\right )} a^{2} +{\left (2 \, d x^{3} + 2 \, c - \sin \left (2 \, d x^{3} + 2 \, c\right )\right )} b^{2} - 8 \, a b \cos \left (d x^{3} + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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