Optimal. Leaf size=96 \[ \frac{2 a x \sin (c+d x)}{d^2}+\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{6 b \sin (c+d x)}{d^4}+\frac{6 b x \cos (c+d x)}{d^3}-\frac{b x^3 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.207823, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6742, 3296, 2638, 2637} \[ \frac{2 a x \sin (c+d x)}{d^2}+\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{6 b \sin (c+d x)}{d^4}+\frac{6 b x \cos (c+d x)}{d^3}-\frac{b x^3 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3296
Rule 2638
Rule 2637
Rubi steps
\begin{align*} \int x^2 (a+b x) \sin (c+d x) \, dx &=\int \left (a x^2 \sin (c+d x)+b x^3 \sin (c+d x)\right ) \, dx\\ &=a \int x^2 \sin (c+d x) \, dx+b \int x^3 \sin (c+d x) \, dx\\ &=-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}+\frac{(2 a) \int x \cos (c+d x) \, dx}{d}+\frac{(3 b) \int x^2 \cos (c+d x) \, dx}{d}\\ &=-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{(2 a) \int \sin (c+d x) \, dx}{d^2}-\frac{(6 b) \int x \sin (c+d x) \, dx}{d^2}\\ &=\frac{2 a \cos (c+d x)}{d^3}+\frac{6 b x \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{(6 b) \int \cos (c+d x) \, dx}{d^3}\\ &=\frac{2 a \cos (c+d x)}{d^3}+\frac{6 b x \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}-\frac{6 b \sin (c+d x)}{d^4}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{3 b x^2 \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.133511, size = 65, normalized size = 0.68 \[ \frac{\left (2 a d^2 x+3 b \left (d^2 x^2-2\right )\right ) \sin (c+d x)-d \left (a \left (d^2 x^2-2\right )+b x \left (d^2 x^2-6\right )\right ) \cos (c+d x)}{d^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 225, normalized size = 2.3 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{b \left ( - \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) +3\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) -6\,\sin \left ( dx+c \right ) +6\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{d}}+a \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) -3\,{\frac{cb \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{d}}-2\,ac \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) +3\,{\frac{{c}^{2}b \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{d}}-a{c}^{2}\cos \left ( dx+c \right ) +{\frac{{c}^{3}b\cos \left ( dx+c \right ) }{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02957, size = 271, normalized size = 2.82 \begin{align*} -\frac{a c^{2} \cos \left (d x + c\right ) - \frac{b c^{3} \cos \left (d x + c\right )}{d} - 2 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c + \frac{3 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{2}}{d} +{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \,{\left (d x + c\right )} \sin \left (d x + c\right )\right )} a - \frac{3 \,{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \,{\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c}{d} + \frac{{\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \,{\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b}{d}}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67678, size = 149, normalized size = 1.55 \begin{align*} -\frac{{\left (b d^{3} x^{3} + a d^{3} x^{2} - 6 \, b d x - 2 \, a d\right )} \cos \left (d x + c\right ) -{\left (3 \, b d^{2} x^{2} + 2 \, a d^{2} x - 6 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18191, size = 117, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{a x^{2} \cos{\left (c + d x \right )}}{d} + \frac{2 a x \sin{\left (c + d x \right )}}{d^{2}} + \frac{2 a \cos{\left (c + d x \right )}}{d^{3}} - \frac{b x^{3} \cos{\left (c + d x \right )}}{d} + \frac{3 b x^{2} \sin{\left (c + d x \right )}}{d^{2}} + \frac{6 b x \cos{\left (c + d x \right )}}{d^{3}} - \frac{6 b \sin{\left (c + d x \right )}}{d^{4}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{3}}{3} + \frac{b x^{4}}{4}\right ) \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09647, size = 92, normalized size = 0.96 \begin{align*} -\frac{{\left (b d^{3} x^{3} + a d^{3} x^{2} - 6 \, b d x - 2 \, a d\right )} \cos \left (d x + c\right )}{d^{4}} + \frac{{\left (3 \, b d^{2} x^{2} + 2 \, a d^{2} x - 6 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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