Optimal. Leaf size=126 \[ \frac{2 a x \sin (c+d x)}{d^2}+\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{5 b x^4 \sin (c+d x)}{d^2}-\frac{60 b x^2 \sin (c+d x)}{d^4}+\frac{20 b x^3 \cos (c+d x)}{d^3}+\frac{120 b \sin (c+d x)}{d^6}-\frac{120 b x \cos (c+d x)}{d^5}-\frac{b x^5 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.19089, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3339, 3296, 2638, 2637} \[ \frac{2 a x \sin (c+d x)}{d^2}+\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{5 b x^4 \sin (c+d x)}{d^2}-\frac{60 b x^2 \sin (c+d x)}{d^4}+\frac{20 b x^3 \cos (c+d x)}{d^3}+\frac{120 b \sin (c+d x)}{d^6}-\frac{120 b x \cos (c+d x)}{d^5}-\frac{b x^5 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3339
Rule 3296
Rule 2638
Rule 2637
Rubi steps
\begin{align*} \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx &=\int \left (a x^2 \sin (c+d x)+b x^5 \sin (c+d x)\right ) \, dx\\ &=a \int x^2 \sin (c+d x) \, dx+b \int x^5 \sin (c+d x) \, dx\\ &=-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^5 \cos (c+d x)}{d}+\frac{(2 a) \int x \cos (c+d x) \, dx}{d}+\frac{(5 b) \int x^4 \cos (c+d x) \, dx}{d}\\ &=-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^5 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{5 b x^4 \sin (c+d x)}{d^2}-\frac{(2 a) \int \sin (c+d x) \, dx}{d^2}-\frac{(20 b) \int x^3 \sin (c+d x) \, dx}{d^2}\\ &=\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{20 b x^3 \cos (c+d x)}{d^3}-\frac{b x^5 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{5 b x^4 \sin (c+d x)}{d^2}-\frac{(60 b) \int x^2 \cos (c+d x) \, dx}{d^3}\\ &=\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{20 b x^3 \cos (c+d x)}{d^3}-\frac{b x^5 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}-\frac{60 b x^2 \sin (c+d x)}{d^4}+\frac{5 b x^4 \sin (c+d x)}{d^2}+\frac{(120 b) \int x \sin (c+d x) \, dx}{d^4}\\ &=\frac{2 a \cos (c+d x)}{d^3}-\frac{120 b x \cos (c+d x)}{d^5}-\frac{a x^2 \cos (c+d x)}{d}+\frac{20 b x^3 \cos (c+d x)}{d^3}-\frac{b x^5 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}-\frac{60 b x^2 \sin (c+d x)}{d^4}+\frac{5 b x^4 \sin (c+d x)}{d^2}+\frac{(120 b) \int \cos (c+d x) \, dx}{d^5}\\ &=\frac{2 a \cos (c+d x)}{d^3}-\frac{120 b x \cos (c+d x)}{d^5}-\frac{a x^2 \cos (c+d x)}{d}+\frac{20 b x^3 \cos (c+d x)}{d^3}-\frac{b x^5 \cos (c+d x)}{d}+\frac{120 b \sin (c+d x)}{d^6}+\frac{2 a x \sin (c+d x)}{d^2}-\frac{60 b x^2 \sin (c+d x)}{d^4}+\frac{5 b x^4 \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.163795, size = 84, normalized size = 0.67 \[ \frac{\left (2 a d^4 x+5 b \left (d^4 x^4-12 d^2 x^2+24\right )\right ) \sin (c+d x)-d \left (a d^2 \left (d^2 x^2-2\right )+b x \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \cos (c+d x)}{d^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 392, normalized size = 3.1 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{b \left ( - \left ( dx+c \right ) ^{5}\cos \left ( dx+c \right ) +5\, \left ( dx+c \right ) ^{4}\sin \left ( dx+c \right ) +20\, \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) -60\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) +120\,\sin \left ( dx+c \right ) -120\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{3}}}-5\,{\frac{cb \left ( - \left ( dx+c \right ) ^{4}\cos \left ( dx+c \right ) +4\, \left ( dx+c \right ) ^{3}\sin \left ( dx+c \right ) +12\, \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) -24\,\cos \left ( dx+c \right ) -24\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{3}}}+10\,{\frac{{c}^{2}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}^{3}}}+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 ) -10\,{\frac{{c}^{3}b \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}^{3}}}-2\,ac \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) +5\,{\frac{b{c}^{4} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{3}}}-a{c}^{2}\cos \left ( dx+c \right ) +{\frac{b{c}^{5}\cos \left ( dx+c \right ) }{{d}^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0415, size = 440, normalized size = 3.49 \begin{align*} -\frac{a c^{2} \cos \left (d x + c\right ) - \frac{b c^{5} \cos \left (d x + c\right )}{d^{3}} - 2 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c + \frac{5 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{4}}{d^{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 )} a - \frac{10 \,{\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^{3}}{d^{3}} + \frac{10 \,{\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 c^{2}}{d^{3}} - \frac{5 \,{\left ({\left ({\left (d x + c\right )}^{4} - 12 \,{\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \,{\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b c}{d^{3}} + \frac{{\left ({\left ({\left (d x + c\right )}^{5} - 20 \,{\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \,{\left ({\left (d x + c\right )}^{4} - 12 \,{\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b}{d^{3}}}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65517, size = 197, normalized size = 1.56 \begin{align*} -\frac{{\left (b d^{5} x^{5} + a d^{5} x^{2} - 20 \, b d^{3} x^{3} - 2 \, a d^{3} + 120 \, b d x\right )} \cos \left (d x + c\right ) -{\left (5 \, b d^{4} x^{4} + 2 \, a d^{4} x - 60 \, b d^{2} x^{2} + 120 \, b\right )} \sin \left (d x + c\right )}{d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.13057, size = 151, normalized size = 1.2 \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^{5} \cos{\left (c + d x \right )}}{d} + \frac{5 b x^{4} \sin{\left (c + d x \right )}}{d^{2}} + \frac{20 b x^{3} \cos{\left (c + d x \right )}}{d^{3}} - \frac{60 b x^{2} \sin{\left (c + d x \right )}}{d^{4}} - \frac{120 b x \cos{\left (c + d x \right )}}{d^{5}} + \frac{120 b \sin{\left (c + d x \right )}}{d^{6}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{3}}{3} + \frac{b x^{6}}{6}\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 = 1.10654, size = 119, normalized size = 0.94 \begin{align*} -\frac{{\left (b d^{5} x^{5} + a d^{5} x^{2} - 20 \, b d^{3} x^{3} - 2 \, a d^{3} + 120 \, b d x\right )} \cos \left (d x + c\right )}{d^{6}} + \frac{{\left (5 \, b d^{4} x^{4} + 2 \, a d^{4} x - 60 \, b d^{2} x^{2} + 120 \, b\right )} \sin \left (d x + c\right )}{d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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