Optimal. Leaf size=111 \[ \frac{2 a x \sin (c+d x)}{d^2}+\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{4 b x^3 \sin (c+d x)}{d^2}+\frac{12 b x^2 \cos (c+d x)}{d^3}-\frac{24 b x \sin (c+d x)}{d^4}-\frac{24 b \cos (c+d x)}{d^5}-\frac{b x^4 \cos (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16329, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3339, 3296, 2638} \[ \frac{2 a x \sin (c+d x)}{d^2}+\frac{2 a \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}+\frac{4 b x^3 \sin (c+d x)}{d^2}+\frac{12 b x^2 \cos (c+d x)}{d^3}-\frac{24 b x \sin (c+d x)}{d^4}-\frac{24 b \cos (c+d x)}{d^5}-\frac{b x^4 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3339
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx &=\int \left (a x^2 \sin (c+d x)+b x^4 \sin (c+d x)\right ) \, dx\\ &=a \int x^2 \sin (c+d x) \, dx+b \int x^4 \sin (c+d x) \, dx\\ &=-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^4 \cos (c+d x)}{d}+\frac{(2 a) \int x \cos (c+d x) \, dx}{d}+\frac{(4 b) \int x^3 \cos (c+d x) \, dx}{d}\\ &=-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^4 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{4 b x^3 \sin (c+d x)}{d^2}-\frac{(2 a) \int \sin (c+d x) \, dx}{d^2}-\frac{(12 b) \int x^2 \sin (c+d x) \, dx}{d^2}\\ &=\frac{2 a \cos (c+d x)}{d^3}+\frac{12 b x^2 \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^4 \cos (c+d x)}{d}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{4 b x^3 \sin (c+d x)}{d^2}-\frac{(24 b) \int x \cos (c+d x) \, dx}{d^3}\\ &=\frac{2 a \cos (c+d x)}{d^3}+\frac{12 b x^2 \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^4 \cos (c+d x)}{d}-\frac{24 b x \sin (c+d x)}{d^4}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{4 b x^3 \sin (c+d x)}{d^2}+\frac{(24 b) \int \sin (c+d x) \, dx}{d^4}\\ &=-\frac{24 b \cos (c+d x)}{d^5}+\frac{2 a \cos (c+d x)}{d^3}+\frac{12 b x^2 \cos (c+d x)}{d^3}-\frac{a x^2 \cos (c+d x)}{d}-\frac{b x^4 \cos (c+d x)}{d}-\frac{24 b x \sin (c+d x)}{d^4}+\frac{2 a x \sin (c+d x)}{d^2}+\frac{4 b x^3 \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.143472, size = 75, normalized size = 0.68 \[ \frac{2 d x \left (a d^2+2 b \left (d^2 x^2-6\right )\right ) \sin (c+d x)-\left (a d^2 \left (d^2 x^2-2\right )+b \left (d^4 x^4-12 d^2 x^2+24\right )\right ) \cos (c+d x)}{d^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 302, normalized size = 2.7 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{b \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}^{2}}}-4\,{\frac{cb \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}}}+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 ) +6\,{\frac{{c}^{2}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}^{2}}}-2\,ac \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) -4\,{\frac{{c}^{3}b \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{2}}}-a{c}^{2}\cos \left ( dx+c \right ) -{\frac{b{c}^{4}\cos \left ( dx+c \right ) }{{d}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.04283, size = 348, normalized size = 3.14 \begin{align*} -\frac{a c^{2} \cos \left (d x + c\right ) + \frac{b c^{4} \cos \left (d x + c\right )}{d^{2}} - 2 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c - \frac{4 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{3}}{d^{2}} +{\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{6 \,{\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^{2}}{d^{2}} - \frac{4 \,{\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}{d^{2}} + \frac{{\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}{d^{2}}}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.35728, size = 171, normalized size = 1.54 \begin{align*} -\frac{{\left (b d^{4} x^{4} - 2 \, a d^{2} +{\left (a d^{4} - 12 \, b d^{2}\right )} x^{2} + 24 \, b\right )} \cos \left (d x + c\right ) - 2 \,{\left (2 \, b d^{3} x^{3} +{\left (a d^{3} - 12 \, b d\right )} x\right )} \sin \left (d x + c\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.9895, size = 134, normalized size = 1.21 \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^{4} \cos{\left (c + d x \right )}}{d} + \frac{4 b x^{3} \sin{\left (c + d x \right )}}{d^{2}} + \frac{12 b x^{2} \cos{\left (c + d x \right )}}{d^{3}} - \frac{24 b x \sin{\left (c + d x \right )}}{d^{4}} - \frac{24 b \cos{\left (c + d x \right )}}{d^{5}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{3}}{3} + \frac{b x^{5}}{5}\right ) \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16785, size = 107, normalized size = 0.96 \begin{align*} -\frac{{\left (b d^{4} x^{4} + a d^{4} x^{2} - 12 \, b d^{2} x^{2} - 2 \, a d^{2} + 24 \, b\right )} \cos \left (d x + c\right )}{d^{5}} + \frac{2 \,{\left (2 \, b d^{3} x^{3} + a d^{3} x - 12 \, b d x\right )} \sin \left (d x + c\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]