Optimal. Leaf size=44 \[ \frac{a x^4}{4}+\frac{b \sin \left (c+d x^2\right )}{2 d^2}-\frac{b x^2 \cos \left (c+d x^2\right )}{2 d} \]
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Rubi [A] time = 0.0425759, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {14, 3379, 3296, 2637} \[ \frac{a x^4}{4}+\frac{b \sin \left (c+d x^2\right )}{2 d^2}-\frac{b x^2 \cos \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3379
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 \left (a+b \sin \left (c+d x^2\right )\right ) \, dx &=\int \left (a x^3+b x^3 \sin \left (c+d x^2\right )\right ) \, dx\\ &=\frac{a x^4}{4}+b \int x^3 \sin \left (c+d x^2\right ) \, dx\\ &=\frac{a x^4}{4}+\frac{1}{2} b \operatorname{Subst}\left (\int x \sin (c+d x) \, dx,x,x^2\right )\\ &=\frac{a x^4}{4}-\frac{b x^2 \cos \left (c+d x^2\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \cos (c+d x) \, dx,x,x^2\right )}{2 d}\\ &=\frac{a x^4}{4}-\frac{b x^2 \cos \left (c+d x^2\right )}{2 d}+\frac{b \sin \left (c+d x^2\right )}{2 d^2}\\ \end{align*}
Mathematica [A] time = 0.0042806, size = 44, normalized size = 1. \[ \frac{a x^4}{4}+\frac{b \sin \left (c+d x^2\right )}{2 d^2}-\frac{b x^2 \cos \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 40, normalized size = 0.9 \begin{align*}{\frac{a{x}^{4}}{4}}+b \left ( -{\frac{{x}^{2}\cos \left ( d{x}^{2}+c \right ) }{2\,d}}+{\frac{\sin \left ( d{x}^{2}+c \right ) }{2\,{d}^{2}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976055, size = 50, normalized size = 1.14 \begin{align*} \frac{1}{4} \, a x^{4} - \frac{{\left (d x^{2} \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right )\right )} b}{2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92282, size = 93, normalized size = 2.11 \begin{align*} \frac{a d^{2} x^{4} - 2 \, b d x^{2} \cos \left (d x^{2} + c\right ) + 2 \, b \sin \left (d x^{2} + c\right )}{4 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.986867, size = 49, normalized size = 1.11 \begin{align*} \begin{cases} \frac{a x^{4}}{4} - \frac{b x^{2} \cos{\left (c + d x^{2} \right )}}{2 d} + \frac{b \sin{\left (c + d x^{2} \right )}}{2 d^{2}} & \text{for}\: d \neq 0 \\\frac{x^{4} \left (a + b \sin{\left (c \right )}\right )}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17592, size = 82, normalized size = 1.86 \begin{align*} \frac{\frac{{\left ({\left (d x^{2} + c\right )}^{2} - 2 \,{\left (d x^{2} + c\right )} c\right )} a}{d} - \frac{2 \,{\left (d x^{2} \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right )\right )} b}{d}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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