Integrand size = 17, antiderivative size = 126 \[ \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx=\frac {2 a \cos (c+d x)}{d^3}-\frac {120 b x \cos (c+d x)}{d^5}-\frac {a x^2 \cos (c+d x)}{d}+\frac {20 b x^3 \cos (c+d x)}{d^3}-\frac {b x^5 \cos (c+d x)}{d}+\frac {120 b \sin (c+d x)}{d^6}+\frac {2 a x \sin (c+d x)}{d^2}-\frac {60 b x^2 \sin (c+d x)}{d^4}+\frac {5 b x^4 \sin (c+d x)}{d^2} \]
2*a*cos(d*x+c)/d^3-120*b*x*cos(d*x+c)/d^5-a*x^2*cos(d*x+c)/d+20*b*x^3*cos( d*x+c)/d^3-b*x^5*cos(d*x+c)/d+120*b*sin(d*x+c)/d^6+2*a*x*sin(d*x+c)/d^2-60 *b*x^2*sin(d*x+c)/d^4+5*b*x^4*sin(d*x+c)/d^2
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67 \[ \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx=\frac {-d \left (a d^2 \left (-2+d^2 x^2\right )+b x \left (120-20 d^2 x^2+d^4 x^4\right )\right ) \cos (c+d x)+\left (2 a d^4 x+5 b \left (24-12 d^2 x^2+d^4 x^4\right )\right ) \sin (c+d x)}{d^6} \]
(-(d*(a*d^2*(-2 + d^2*x^2) + b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Cos[c + d*x ]) + (2*a*d^4*x + 5*b*(24 - 12*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^6
Time = 0.37 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3820, 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.
| \(\displaystyle \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx\) |
| \(\Big \downarrow \) 3820 |
| \(\displaystyle \int \left (a x^2 \sin (c+d x)+b x^5 \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a \cos (c+d x)}{d^3}+\frac {2 a x \sin (c+d x)}{d^2}-\frac {a x^2 \cos (c+d x)}{d}+\frac {120 b \sin (c+d x)}{d^6}-\frac {120 b x \cos (c+d x)}{d^5}-\frac {60 b x^2 \sin (c+d x)}{d^4}+\frac {20 b x^3 \cos (c+d x)}{d^3}+\frac {5 b x^4 \sin (c+d x)}{d^2}-\frac {b x^5 \cos (c+d x)}{d}\) |
(2*a*Cos[c + d*x])/d^3 - (120*b*x*Cos[c + d*x])/d^5 - (a*x^2*Cos[c + d*x]) /d + (20*b*x^3*Cos[c + d*x])/d^3 - (b*x^5*Cos[c + d*x])/d + (120*b*Sin[c + d*x])/d^6 + (2*a*x*Sin[c + d*x])/d^2 - (60*b*x^2*Sin[c + d*x])/d^4 + (5*b *x^4*Sin[c + d*x])/d^2
3.1.80.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_ )], x_Symbol] :> Int[ExpandIntegrand[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x ], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {\left (b \,d^{4} x^{5}+a \,d^{4} x^{2}-20 b \,d^{2} x^{3}-2 a \,d^{2}+120 b x \right ) \cos \left (d x +c \right )}{d^{5}}+\frac {\left (5 b \,x^{4} d^{4}+2 a \,d^{4} x -60 d^{2} x^{2} b +120 b \right ) \sin \left (d x +c \right )}{d^{6}}\) | \(88\) |
| parallelrisch | \(\frac {\left (x \left (b \,x^{3}+a \right ) d^{4}-20 d^{2} x^{2} b +120 b \right ) x d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\left (10 b \,x^{4}+4 a x \right ) d^{4}-120 d^{2} x^{2} b +240 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (x^{2} \left (b \,x^{3}+a \right ) d^{4}+\left (-20 b \,x^{3}-4 a \right ) d^{2}+120 b x \right ) d}{d^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(134\) |
| norman | \(\frac {\frac {4 a}{d^{3}}+\frac {a \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {b \,x^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \,x^{2}}{d}+\frac {240 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{6}}-\frac {120 b x}{d^{5}}+\frac {20 b \,x^{3}}{d^{3}}-\frac {b \,x^{5}}{d}+\frac {4 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}+\frac {120 b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{5}}-\frac {120 b \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}-\frac {20 b \,x^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{3}}+\frac {10 b \,x^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\) | \(201\) |
| meijerg | \(\frac {32 b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {15}{4 \sqrt {\pi }}+\frac {\left (\frac {15}{8} d^{4} x^{4}-\frac {45}{2} d^{2} x^{2}+45\right ) \cos \left (d x \right )}{12 \sqrt {\pi }}+\frac {x d \left (\frac {3}{8} d^{4} x^{4}-\frac {15}{2} d^{2} x^{2}+45\right ) \sin \left (d x \right )}{12 \sqrt {\pi }}\right )}{d^{6}}+\frac {32 b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {x d \left (\frac {7}{8} d^{4} x^{4}-\frac {35}{2} d^{2} x^{2}+105\right ) \cos \left (d x \right )}{28 \sqrt {\pi }}+\frac {\left (\frac {35}{8} d^{4} x^{4}-\frac {105}{2} d^{2} x^{2}+105\right ) \sin \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{6}}+\frac {4 a \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {x \left (d^{2}\right )^{\frac {3}{2}} \cos \left (d x \right )}{2 \sqrt {\pi }\, d^{2}}-\frac {\left (d^{2}\right )^{\frac {3}{2}} \left (-\frac {3 d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{6 \sqrt {\pi }\, d^{3}}\right )}{d^{2} \sqrt {d^{2}}}+\frac {4 a \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {d^{2} x^{2}}{2}+1\right ) \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {d x \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\) | \(252\) |
| parts | \(-\frac {b \,x^{5} \cos \left (d x +c \right )}{d}-\frac {a \,x^{2} \cos \left (d x +c \right )}{d}+\frac {-\frac {2 a c \sin \left (d x +c \right )}{d}+\frac {2 a \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}+\frac {5 b \,c^{4} \sin \left (d x +c \right )}{d^{4}}-\frac {20 b \,c^{3} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}+\frac {30 b \,c^{2} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}-\frac {20 b c \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}+\frac {5 b \left (\left (d x +c \right )^{4} \sin \left (d x +c \right )+4 \left (d x +c \right )^{3} \cos \left (d x +c \right )-12 \left (d x +c \right )^{2} \sin \left (d x +c \right )+24 \sin \left (d x +c \right )-24 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{4}}}{d^{2}}\) | \(293\) |
| derivativedivides | \(\frac {-a \,c^{2} \cos \left (d x +c \right )-2 a c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+\frac {b \,c^{5} \cos \left (d x +c \right )}{d^{3}}+\frac {5 b \,c^{4} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {10 b \,c^{3} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {10 b \,c^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {5 b c \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}}{d^{3}}\) | \(392\) |
| default | \(\frac {-a \,c^{2} \cos \left (d x +c \right )-2 a c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+\frac {b \,c^{5} \cos \left (d x +c \right )}{d^{3}}+\frac {5 b \,c^{4} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {10 b \,c^{3} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {10 b \,c^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {5 b c \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}}{d^{3}}\) | \(392\) |
-1/d^5*(b*d^4*x^5+a*d^4*x^2-20*b*d^2*x^3-2*a*d^2+120*b*x)*cos(d*x+c)+(5*b* d^4*x^4+2*a*d^4*x-60*b*d^2*x^2+120*b)/d^6*sin(d*x+c)
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69 \[ \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx=-\frac {{\left (b d^{5} x^{5} + a d^{5} x^{2} - 20 \, b d^{3} x^{3} - 2 \, a d^{3} + 120 \, b d x\right )} \cos \left (d x + c\right ) - {\left (5 \, b d^{4} x^{4} + 2 \, a d^{4} x - 60 \, b d^{2} x^{2} + 120 \, b\right )} \sin \left (d x + c\right )}{d^{6}} \]
-((b*d^5*x^5 + a*d^5*x^2 - 20*b*d^3*x^3 - 2*a*d^3 + 120*b*d*x)*cos(d*x + c ) - (5*b*d^4*x^4 + 2*a*d^4*x - 60*b*d^2*x^2 + 120*b)*sin(d*x + c))/d^6
Time = 0.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.20 \[ \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx=\begin {cases} - \frac {a x^{2} \cos {\left (c + d x \right )}}{d} + \frac {2 a x \sin {\left (c + d x \right )}}{d^{2}} + \frac {2 a \cos {\left (c + d x \right )}}{d^{3}} - \frac {b x^{5} \cos {\left (c + d x \right )}}{d} + \frac {5 b x^{4} \sin {\left (c + d x \right )}}{d^{2}} + \frac {20 b x^{3} \cos {\left (c + d x \right )}}{d^{3}} - \frac {60 b x^{2} \sin {\left (c + d x \right )}}{d^{4}} - \frac {120 b x \cos {\left (c + d x \right )}}{d^{5}} + \frac {120 b \sin {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{3}}{3} + \frac {b x^{6}}{6}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \]
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**5*cos(c + d*x)/d + 5*b*x**4*sin(c + d*x)/d**2 + 20*b*x**3 *cos(c + d*x)/d**3 - 60*b*x**2*sin(c + d*x)/d**4 - 120*b*x*cos(c + d*x)/d* *5 + 120*b*sin(c + d*x)/d**6, Ne(d, 0)), ((a*x**3/3 + b*x**6/6)*sin(c), Tr ue))
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (126) = 252\).
Time = 0.21 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.59 \[ \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx=-\frac {a c^{2} \cos \left (d x + c\right ) - \frac {b c^{5} \cos \left (d x + c\right )}{d^{3}} - 2 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c + \frac {5 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{4}}{d^{3}} + {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a - \frac {10 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c^{3}}{d^{3}} + \frac {10 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b c^{2}}{d^{3}} - \frac {5 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b c}{d^{3}} + \frac {{\left ({\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \, {\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b}{d^{3}}}{d^{3}} \]
-(a*c^2*cos(d*x + c) - b*c^5*cos(d*x + c)/d^3 - 2*((d*x + c)*cos(d*x + c) - sin(d*x + c))*a*c + 5*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b*c^4/d^3 + (((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a - 10*(((d* x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b*c^3/d^3 + 10*(((d *x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))* b*c^2/d^3 - 5*(((d*x + c)^4 - 12*(d*x + c)^2 + 24)*cos(d*x + c) - 4*((d*x + c)^3 - 6*d*x - 6*c)*sin(d*x + c))*b*c/d^3 + (((d*x + c)^5 - 20*(d*x + c) ^3 + 120*d*x + 120*c)*cos(d*x + c) - 5*((d*x + c)^4 - 12*(d*x + c)^2 + 24) *sin(d*x + c))*b/d^3)/d^3
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx=-\frac {{\left (b d^{5} x^{5} + a d^{5} x^{2} - 20 \, b d^{3} x^{3} - 2 \, a d^{3} + 120 \, b d x\right )} \cos \left (d x + c\right )}{d^{6}} + \frac {{\left (5 \, b d^{4} x^{4} + 2 \, a d^{4} x - 60 \, b d^{2} x^{2} + 120 \, b\right )} \sin \left (d x + c\right )}{d^{6}} \]
-(b*d^5*x^5 + a*d^5*x^2 - 20*b*d^3*x^3 - 2*a*d^3 + 120*b*d*x)*cos(d*x + c) /d^6 + (5*b*d^4*x^4 + 2*a*d^4*x - 60*b*d^2*x^2 + 120*b)*sin(d*x + c)/d^6
Time = 6.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int x^2 \left (a+b x^3\right ) \sin (c+d x) \, dx=\frac {120\,b\,\sin \left (c+d\,x\right )+d^4\,\left (5\,b\,x^4\,\sin \left (c+d\,x\right )+2\,a\,x\,\sin \left (c+d\,x\right )\right )-d^5\,\left (a\,x^2\,\cos \left (c+d\,x\right )+b\,x^5\,\cos \left (c+d\,x\right )\right )+d^3\,\left (2\,a\,\cos \left (c+d\,x\right )+20\,b\,x^3\,\cos \left (c+d\,x\right )\right )-60\,b\,d^2\,x^2\,\sin \left (c+d\,x\right )-120\,b\,d\,x\,\cos \left (c+d\,x\right )}{d^6} \]