Optimal. Leaf size=58 \[ \frac{1}{4} x^2 \left (2 a^2+b^2\right )-\frac{a b \cos \left (c+d x^2\right )}{d}-\frac{b^2 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{4 d} \]
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Rubi [A] time = 0.0485753, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3379, 2644} \[ \frac{1}{4} x^2 \left (2 a^2+b^2\right )-\frac{a b \cos \left (c+d x^2\right )}{d}-\frac{b^2 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 2644
Rubi steps
\begin{align*} \int x \left (a+b \sin \left (c+d x^2\right )\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b \sin (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac{1}{4} \left (2 a^2+b^2\right ) x^2-\frac{a b \cos \left (c+d x^2\right )}{d}-\frac{b^2 \cos \left (c+d x^2\right ) \sin \left (c+d x^2\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.123084, size = 52, normalized size = 0.9 \[ -\frac{-2 \left (2 a^2+b^2\right ) \left (c+d x^2\right )+8 a b \cos \left (c+d x^2\right )+b^2 \sin \left (2 \left (c+d x^2\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 62, normalized size = 1.1 \begin{align*}{\frac{1}{2\,d} \left ({b}^{2} \left ( -{\frac{\cos \left ( d{x}^{2}+c \right ) \sin \left ( d{x}^{2}+c \right ) }{2}}+{\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ) -2\,ab\cos \left ( d{x}^{2}+c \right ) +{a}^{2} \left ( d{x}^{2}+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984389, size = 70, normalized size = 1.21 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + \frac{{\left (2 \, d x^{2} - \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} b^{2}}{8 \, d} - \frac{a b \cos \left (d x^{2} + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04341, size = 119, normalized size = 2.05 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )} d x^{2} - b^{2} \cos \left (d x^{2} + c\right ) \sin \left (d x^{2} + c\right ) - 4 \, a b \cos \left (d x^{2} + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.607337, size = 95, normalized size = 1.64 \begin{align*} \begin{cases} \frac{a^{2} x^{2}}{2} - \frac{a b \cos{\left (c + d x^{2} \right )}}{d} + \frac{b^{2} x^{2} \sin ^{2}{\left (c + d x^{2} \right )}}{4} + \frac{b^{2} x^{2} \cos ^{2}{\left (c + d x^{2} \right )}}{4} - \frac{b^{2} \sin{\left (c + d x^{2} \right )} \cos{\left (c + d x^{2} \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x^{2} \left (a + b \sin{\left (c \right )}\right )^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11968, size = 77, normalized size = 1.33 \begin{align*} \frac{4 \,{\left (d x^{2} + c\right )} a^{2} +{\left (2 \, d x^{2} + 2 \, c - \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} b^{2} - 8 \, a b \cos \left (d x^{2} + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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