Optimal. Leaf size=80 \[ \frac{a \sin (c+d x)}{d^2}-\frac{a x \cos (c+d x)}{d}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{6 b \sin (c+d x)}{d^4}+\frac{6 b x \cos (c+d x)}{d^3}-\frac{b x^3 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.101841, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3339, 3296, 2637} \[ \frac{a \sin (c+d x)}{d^2}-\frac{a x \cos (c+d x)}{d}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{6 b \sin (c+d x)}{d^4}+\frac{6 b x \cos (c+d x)}{d^3}-\frac{b x^3 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3339
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x \left (a+b x^2\right ) \sin (c+d x) \, dx &=\int \left (a x \sin (c+d x)+b x^3 \sin (c+d x)\right ) \, dx\\ &=a \int x \sin (c+d x) \, dx+b \int x^3 \sin (c+d x) \, dx\\ &=-\frac{a x \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}+\frac{a \int \cos (c+d x) \, dx}{d}+\frac{(3 b) \int x^2 \cos (c+d x) \, dx}{d}\\ &=-\frac{a x \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}+\frac{a \sin (c+d x)}{d^2}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{(6 b) \int x \sin (c+d x) \, dx}{d^2}\\ &=\frac{6 b x \cos (c+d x)}{d^3}-\frac{a x \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}+\frac{a \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{6 b x \cos (c+d x)}{d^3}-\frac{a x \cos (c+d x)}{d}-\frac{b x^3 \cos (c+d x)}{d}-\frac{6 b \sin (c+d x)}{d^4}+\frac{a \sin (c+d x)}{d^2}+\frac{3 b x^2 \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.111948, size = 57, normalized size = 0.71 \[ \frac{\left (a d^2+3 b \left (d^2 x^2-2\right )\right ) \sin (c+d x)-d x \left (a d^2+b \left (d^2 x^2-6\right )\right ) \cos (c+d x)}{d^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 181, normalized size = 2.3 \begin{align*}{\frac{1}{{d}^{2}} \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}^{2}}}-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}}}+a \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}^{2}}}+ac\cos \left ( dx+c \right ) +{\frac{{c}^{3}b\cos \left ( dx+c \right ) }{{d}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02718, size = 223, normalized size = 2.79 \begin{align*} \frac{a c \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 - \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^{2}} + \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^{2}} - \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^{2}}}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34065, size = 130, normalized size = 1.62 \begin{align*} -\frac{{\left (b d^{3} x^{3} +{\left (a d^{3} - 6 \, b d\right )} x\right )} \cos \left (d x + c\right ) -{\left (3 \, b d^{2} x^{2} + a d^{2} - 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 = 1.43142, size = 99, normalized size = 1.24 \begin{align*} \begin{cases} - \frac{a x \cos{\left (c + d x \right )}}{d} + \frac{a \sin{\left (c + d x \right )}}{d^{2}} - \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^{2}}{2} + \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 = 1.09518, size = 81, normalized size = 1.01 \begin{align*} -\frac{{\left (b d^{3} x^{3} + a d^{3} x - 6 \, b d x\right )} \cos \left (d x + c\right )}{d^{4}} + \frac{{\left (3 \, b d^{2} x^{2} + a d^{2} - 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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