Integrand size = 15, antiderivative size = 96 \[ \int x^2 (a+b x) \sin (c+d x) \, dx=\frac {2 a \cos (c+d x)}{d^3}+\frac {6 b x \cos (c+d x)}{d^3}-\frac {a x^2 \cos (c+d x)}{d}-\frac {b x^3 \cos (c+d x)}{d}-\frac {6 b \sin (c+d x)}{d^4}+\frac {2 a x \sin (c+d x)}{d^2}+\frac {3 b x^2 \sin (c+d x)}{d^2} \]
2*a*cos(d*x+c)/d^3+6*b*x*cos(d*x+c)/d^3-a*x^2*cos(d*x+c)/d-b*x^3*cos(d*x+c )/d-6*b*sin(d*x+c)/d^4+2*a*x*sin(d*x+c)/d^2+3*b*x^2*sin(d*x+c)/d^2
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int x^2 (a+b x) \sin (c+d x) \, dx=\frac {-d \left (b x \left (-6+d^2 x^2\right )+a \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+\left (2 a d^2 x+3 b \left (-2+d^2 x^2\right )\right ) \sin (c+d x)}{d^4} \]
(-(d*(b*x*(-6 + d^2*x^2) + a*(-2 + d^2*x^2))*Cos[c + d*x]) + (2*a*d^2*x + 3*b*(-2 + d^2*x^2))*Sin[c + d*x])/d^4
Time = 0.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 (a+b x) \sin (c+d x) \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (a x^2 \sin (c+d x)+b x^3 \sin (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a \cos (c+d x)}{d^3}+\frac {2 a x \sin (c+d x)}{d^2}-\frac {a x^2 \cos (c+d x)}{d}-\frac {6 b \sin (c+d x)}{d^4}+\frac {6 b x \cos (c+d x)}{d^3}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {b x^3 \cos (c+d x)}{d}\) |
(2*a*Cos[c + d*x])/d^3 + (6*b*x*Cos[c + d*x])/d^3 - (a*x^2*Cos[c + d*x])/d - (b*x^3*Cos[c + d*x])/d - (6*b*Sin[c + d*x])/d^4 + (2*a*x*Sin[c + d*x])/ d^2 + (3*b*x^2*Sin[c + d*x])/d^2
3.1.2.3.1 Defintions of rubi rules used
Time = 0.14 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {\left (b \,d^{2} x^{3}+a \,d^{2} x^{2}-6 b x -2 a \right ) \cos \left (d x +c \right )}{d^{3}}+\frac {\left (3 d^{2} x^{2} b +2 a \,d^{2} x -6 b \right ) \sin \left (d x +c \right )}{d^{4}}\) | \(67\) |
parallelrisch | \(\frac {x \left (x \left (b x +a \right ) d^{2}-6 b \right ) d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\left (6 b \,x^{2}+4 a x \right ) d^{2}-12 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (x^{2} \left (b x +a \right ) d^{2}-6 b x -4 a \right ) d}{d^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(101\) |
norman | \(\frac {\frac {4 a}{d^{3}}+\frac {a \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {b \,x^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \,x^{2}}{d}-\frac {12 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}+\frac {6 b x}{d^{3}}-\frac {b \,x^{3}}{d}+\frac {4 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}-\frac {6 b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{3}}+\frac {6 b \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\) | \(154\) |
parts | \(-\frac {b \,x^{3} \cos \left (d x +c \right )}{d}-\frac {a \,x^{2} \cos \left (d x +c \right )}{d}+\frac {-\frac {2 a c \sin \left (d x +c \right )}{d}+\frac {2 a \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}+\frac {3 b \,c^{2} \sin \left (d x +c \right )}{d^{2}}-\frac {6 b c \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}+\frac {3 b \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}}{d^{2}}\) | \(158\) |
meijerg | \(\frac {8 b \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 d^{2} x^{2}}{2}+3\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}-\frac {d x \left (-\frac {d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}+\frac {8 b \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {x d \left (-\frac {5 d^{2} x^{2}}{2}+15\right ) \cos \left (d x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 d^{2} x^{2}}{2}+15\right ) \sin \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}+\frac {4 a \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {x \left (d^{2}\right )^{\frac {3}{2}} \cos \left (d x \right )}{2 \sqrt {\pi }\, d^{2}}-\frac {\left (d^{2}\right )^{\frac {3}{2}} \left (-\frac {3 d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{6 \sqrt {\pi }\, d^{3}}\right )}{d^{2} \sqrt {d^{2}}}+\frac {4 a \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {d^{2} x^{2}}{2}+1\right ) \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {d x \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\) | \(220\) |
derivativedivides | \(\frac {-a \,c^{2} \cos \left (d x +c \right )-2 a c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+\frac {b \,c^{3} \cos \left (d x +c \right )}{d}+\frac {3 b \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}-\frac {3 b c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}+\frac {b \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}}{d^{3}}\) | \(225\) |
default | \(\frac {-a \,c^{2} \cos \left (d x +c \right )-2 a c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+\frac {b \,c^{3} \cos \left (d x +c \right )}{d}+\frac {3 b \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}-\frac {3 b c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}+\frac {b \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}}{d^{3}}\) | \(225\) |
-1/d^3*(b*d^2*x^3+a*d^2*x^2-6*b*x-2*a)*cos(d*x+c)+(3*b*d^2*x^2+2*a*d^2*x-6 *b)/d^4*sin(d*x+c)
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.70 \[ \int x^2 (a+b x) \sin (c+d x) \, dx=-\frac {{\left (b d^{3} x^{3} + a d^{3} x^{2} - 6 \, b d x - 2 \, a d\right )} \cos \left (d x + c\right ) - {\left (3 \, b d^{2} x^{2} + 2 \, a d^{2} x - 6 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \]
-((b*d^3*x^3 + a*d^3*x^2 - 6*b*d*x - 2*a*d)*cos(d*x + c) - (3*b*d^2*x^2 + 2*a*d^2*x - 6*b)*sin(d*x + c))/d^4
Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.22 \[ \int x^2 (a+b x) \sin (c+d x) \, dx=\begin {cases} - \frac {a x^{2} \cos {\left (c + d x \right )}}{d} + \frac {2 a x \sin {\left (c + d x \right )}}{d^{2}} + \frac {2 a \cos {\left (c + d x \right )}}{d^{3}} - \frac {b x^{3} \cos {\left (c + d x \right )}}{d} + \frac {3 b x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {6 b x \cos {\left (c + d x \right )}}{d^{3}} - \frac {6 b \sin {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{3}}{3} + \frac {b x^{4}}{4}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((-a*x**2*cos(c + d*x)/d + 2*a*x*sin(c + d*x)/d**2 + 2*a*cos(c + d*x)/d**3 - b*x**3*cos(c + d*x)/d + 3*b*x**2*sin(c + d*x)/d**2 + 6*b*x*cos (c + d*x)/d**3 - 6*b*sin(c + d*x)/d**4, Ne(d, 0)), ((a*x**3/3 + b*x**4/4)* sin(c), True))
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (96) = 192\).
Time = 0.21 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.09 \[ \int x^2 (a+b x) \sin (c+d x) \, dx=-\frac {a c^{2} \cos \left (d x + c\right ) - \frac {b c^{3} \cos \left (d x + c\right )}{d} - 2 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c + \frac {3 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{2}}{d} + {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a - \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c}{d} + \frac {{\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b}{d}}{d^{3}} \]
-(a*c^2*cos(d*x + c) - b*c^3*cos(d*x + c)/d - 2*((d*x + c)*cos(d*x + c) - sin(d*x + c))*a*c + 3*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b*c^2/d + (( (d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a - 3*(((d*x + c )^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b*c/d + (((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*b/d)/d^3
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.71 \[ \int x^2 (a+b x) \sin (c+d x) \, dx=-\frac {{\left (b d^{3} x^{3} + a d^{3} x^{2} - 6 \, b d x - 2 \, a d\right )} \cos \left (d x + c\right )}{d^{4}} + \frac {{\left (3 \, b d^{2} x^{2} + 2 \, a d^{2} x - 6 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \]
-(b*d^3*x^3 + a*d^3*x^2 - 6*b*d*x - 2*a*d)*cos(d*x + c)/d^4 + (3*b*d^2*x^2 + 2*a*d^2*x - 6*b)*sin(d*x + c)/d^4
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int x^2 (a+b x) \sin (c+d x) \, dx=\frac {3\,b\,x^2\,\sin \left (c+d\,x\right )+2\,a\,x\,\sin \left (c+d\,x\right )}{d^2}+\frac {2\,a\,\cos \left (c+d\,x\right )+6\,b\,x\,\cos \left (c+d\,x\right )}{d^3}-\frac {a\,x^2\,\cos \left (c+d\,x\right )+b\,x^3\,\cos \left (c+d\,x\right )}{d}-\frac {6\,b\,\sin \left (c+d\,x\right )}{d^4} \]