Integrand size = 16, antiderivative size = 188 \[ \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx=\frac {720 b^2 \cos (c+d x)}{d^7}-\frac {a^2 \cos (c+d x)}{d}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {720 b^2 x \sin (c+d x)}{d^6}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2} \]
720*b^2*cos(d*x+c)/d^7-a^2*cos(d*x+c)/d+12*a*b*x*cos(d*x+c)/d^3-360*b^2*x^ 2*cos(d*x+c)/d^5-2*a*b*x^3*cos(d*x+c)/d+30*b^2*x^4*cos(d*x+c)/d^3-b^2*x^6* cos(d*x+c)/d-12*a*b*sin(d*x+c)/d^4+720*b^2*x*sin(d*x+c)/d^6+6*a*b*x^2*sin( d*x+c)/d^2-120*b^2*x^3*sin(d*x+c)/d^4+6*b^2*x^5*sin(d*x+c)/d^2
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.60 \[ \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx=\frac {-\left (\left (a^2 d^6+2 a b d^4 x \left (-6+d^2 x^2\right )+b^2 \left (-720+360 d^2 x^2-30 d^4 x^4+d^6 x^6\right )\right ) \cos (c+d x)\right )+6 b d \left (a d^2 \left (-2+d^2 x^2\right )+b x \left (120-20 d^2 x^2+d^4 x^4\right )\right ) \sin (c+d x)}{d^7} \]
(-((a^2*d^6 + 2*a*b*d^4*x*(-6 + d^2*x^2) + b^2*(-720 + 360*d^2*x^2 - 30*d^ 4*x^4 + d^6*x^6))*Cos[c + d*x]) + 6*b*d*(a*d^2*(-2 + d^2*x^2) + b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^7
Time = 0.46 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3810, 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.
| \(\displaystyle \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx\) |
| \(\Big \downarrow \) 3810 |
| \(\displaystyle \int \left (a^2 \sin (c+d x)+2 a b x^3 \sin (c+d x)+b^2 x^6 \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {12 a b x \cos (c+d x)}{d^3}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {720 b^2 \cos (c+d x)}{d^7}+\frac {720 b^2 x \sin (c+d x)}{d^6}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {b^2 x^6 \cos (c+d x)}{d}\) |
(720*b^2*Cos[c + d*x])/d^7 - (a^2*Cos[c + d*x])/d + (12*a*b*x*Cos[c + d*x] )/d^3 - (360*b^2*x^2*Cos[c + d*x])/d^5 - (2*a*b*x^3*Cos[c + d*x])/d + (30* b^2*x^4*Cos[c + d*x])/d^3 - (b^2*x^6*Cos[c + d*x])/d - (12*a*b*Sin[c + d*x ])/d^4 + (720*b^2*x*Sin[c + d*x])/d^6 + (6*a*b*x^2*Sin[c + d*x])/d^2 - (12 0*b^2*x^3*Sin[c + d*x])/d^4 + (6*b^2*x^5*Sin[c + d*x])/d^2
3.1.88.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> In t[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {\left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}-30 b^{2} x^{4} d^{4}+a^{2} d^{6}-12 a b \,d^{4} x +360 d^{2} x^{2} b^{2}-720 b^{2}\right ) \cos \left (d x +c \right )}{d^{7}}+\frac {6 b \left (b \,d^{4} x^{5}+a \,d^{4} x^{2}-20 b \,d^{2} x^{3}-2 a \,d^{2}+120 b x \right ) \sin \left (d x +c \right )}{d^{6}}\) | \(124\) |
| parallelrisch | \(\frac {2 \left (x^{2} \left (\frac {b \,x^{3}}{2}+a \right ) d^{4}+\left (-15 b \,x^{3}-6 a \right ) d^{2}+180 b x \right ) x \,d^{2} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 d b \left (x^{2} \left (b \,x^{3}+a \right ) d^{4}+\left (-20 b \,x^{3}-2 a \right ) d^{2}+120 b x \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-b^{2} x^{6}-2 a b \,x^{3}-2 a^{2}\right ) d^{6}+\left (30 b^{2} x^{4}+12 a b x \right ) d^{4}-360 d^{2} x^{2} b^{2}+1440 b^{2}}{d^{7} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(178\) |
| norman | \(\frac {\frac {b^{2} x^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{2} d^{6}-1440 b^{2}}{d^{7}}-\frac {360 b^{2} x^{2}}{d^{5}}+\frac {30 b^{2} x^{4}}{d^{3}}-\frac {b^{2} x^{6}}{d}-\frac {24 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}+\frac {1440 b^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{6}}+\frac {360 b^{2} x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{5}}-\frac {240 b^{2} x^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}-\frac {30 b^{2} x^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{3}}+\frac {12 b^{2} x^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}+\frac {12 a b x}{d^{3}}-\frac {2 a b \,x^{3}}{d}-\frac {12 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{3}}+\frac {12 a b \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}+\frac {2 a b \,x^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\) | \(286\) |
| meijerg | \(\frac {64 b^{2} \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {x \left (d^{2}\right )^{\frac {7}{2}} \left (\frac {21}{8} d^{4} x^{4}-\frac {105}{2} d^{2} x^{2}+315\right ) \cos \left (d x \right )}{28 \sqrt {\pi }\, d^{6}}-\frac {\left (d^{2}\right )^{\frac {7}{2}} \left (-\frac {7}{16} d^{6} x^{6}+\frac {105}{8} d^{4} x^{4}-\frac {315}{2} d^{2} x^{2}+315\right ) \sin \left (d x \right )}{28 \sqrt {\pi }\, d^{7}}\right )}{d^{6} \sqrt {d^{2}}}+\frac {64 b^{2} \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {45}{4 \sqrt {\pi }}+\frac {\left (-\frac {1}{16} d^{6} x^{6}+\frac {15}{8} d^{4} x^{4}-\frac {45}{2} d^{2} x^{2}+45\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}+\frac {x d \left (\frac {3}{8} d^{4} x^{4}-\frac {15}{2} d^{2} x^{2}+45\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{7}}+\frac {16 b a \sqrt {\pi }\, \sin \left (c \right ) \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 d^{2} x^{2}}{2}+3\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}-\frac {d x \left (-\frac {d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}+\frac {16 b a \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {x d \left (-\frac {5 d^{2} x^{2}}{2}+15\right ) \cos \left (d x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 d^{2} x^{2}}{2}+15\right ) \sin \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}+\frac {a^{2} \sin \left (c \right ) \sin \left (d x \right )}{d}+\frac {a^{2} \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (d x \right )}{\sqrt {\pi }}\right )}{d}\) | \(332\) |
| parts | \(-\frac {b^{2} x^{6} \cos \left (d x +c \right )}{d}-\frac {2 a b \,x^{3} \cos \left (d x +c \right )}{d}-\frac {a^{2} \cos \left (d x +c \right )}{d}+\frac {6 b \left (a \,c^{2} \sin \left (d x +c \right )-2 a c \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+a \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )-\frac {b \,c^{5} \sin \left (d x +c \right )}{d^{3}}+\frac {5 b \,c^{4} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sin \left (d x +c \right )+4 \left (d x +c \right )^{3} \cos \left (d x +c \right )-12 \left (d x +c \right )^{2} \sin \left (d x +c \right )+24 \sin \left (d x +c \right )-24 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {b \left (\left (d x +c \right )^{5} \sin \left (d x +c \right )+5 \left (d x +c \right )^{4} \cos \left (d x +c \right )-20 \left (d x +c \right )^{3} \sin \left (d x +c \right )-60 \left (d x +c \right )^{2} \cos \left (d x +c \right )+120 \cos \left (d x +c \right )+120 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}\right )}{d^{4}}\) | \(435\) |
| derivativedivides | \(\frac {-a^{2} \cos \left (d x +c \right )+\frac {2 a b \,c^{3} \cos \left (d x +c \right )}{d^{3}}+\frac {6 a b \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {6 a 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^{3}}+\frac {2 a 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^{3}}-\frac {b^{2} c^{6} \cos \left (d x +c \right )}{d^{6}}-\frac {6 b^{2} c^{5} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}+\frac {15 b^{2} c^{4} \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^{6}}-\frac {20 b^{2} c^{3} \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^{6}}+\frac {15 b^{2} c^{2} \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^{6}}-\frac {6 b^{2} c \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^{6}}+\frac {b^{2} \left (-\left (d x +c \right )^{6} \cos \left (d x +c \right )+6 \left (d x +c \right )^{5} \sin \left (d x +c \right )+30 \left (d x +c \right )^{4} \cos \left (d x +c \right )-120 \left (d x +c \right )^{3} \sin \left (d x +c \right )-360 \left (d x +c \right )^{2} \cos \left (d x +c \right )+720 \cos \left (d x +c \right )+720 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}}{d}\) | \(599\) |
| default | \(\frac {-a^{2} \cos \left (d x +c \right )+\frac {2 a b \,c^{3} \cos \left (d x +c \right )}{d^{3}}+\frac {6 a b \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}-\frac {6 a 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^{3}}+\frac {2 a 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^{3}}-\frac {b^{2} c^{6} \cos \left (d x +c \right )}{d^{6}}-\frac {6 b^{2} c^{5} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{6}}+\frac {15 b^{2} c^{4} \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^{6}}-\frac {20 b^{2} c^{3} \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^{6}}+\frac {15 b^{2} c^{2} \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^{6}}-\frac {6 b^{2} c \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^{6}}+\frac {b^{2} \left (-\left (d x +c \right )^{6} \cos \left (d x +c \right )+6 \left (d x +c \right )^{5} \sin \left (d x +c \right )+30 \left (d x +c \right )^{4} \cos \left (d x +c \right )-120 \left (d x +c \right )^{3} \sin \left (d x +c \right )-360 \left (d x +c \right )^{2} \cos \left (d x +c \right )+720 \cos \left (d x +c \right )+720 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}}{d}\) | \(599\) |
-(b^2*d^6*x^6+2*a*b*d^6*x^3-30*b^2*d^4*x^4+a^2*d^6-12*a*b*d^4*x+360*b^2*d^ 2*x^2-720*b^2)/d^7*cos(d*x+c)+6*b/d^6*(b*d^4*x^5+a*d^4*x^2-20*b*d^2*x^3-2* a*d^2+120*b*x)*sin(d*x+c)
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.69 \[ \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx=-\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} - 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right ) - 6 \, {\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \]
-((b^2*d^6*x^6 + 2*a*b*d^6*x^3 - 30*b^2*d^4*x^4 + a^2*d^6 - 12*a*b*d^4*x + 360*b^2*d^2*x^2 - 720*b^2)*cos(d*x + c) - 6*(b^2*d^5*x^5 + a*b*d^5*x^2 - 20*b^2*d^3*x^3 - 2*a*b*d^3 + 120*b^2*d*x)*sin(d*x + c))/d^7
Time = 0.57 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.20 \[ \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx=\begin {cases} - \frac {a^{2} \cos {\left (c + d x \right )}}{d} - \frac {2 a b x^{3} \cos {\left (c + d x \right )}}{d} + \frac {6 a b x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \cos {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \sin {\left (c + d x \right )}}{d^{4}} - \frac {b^{2} x^{6} \cos {\left (c + d x \right )}}{d} + \frac {6 b^{2} x^{5} \sin {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \cos {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \sin {\left (c + d x \right )}}{d^{4}} - \frac {360 b^{2} x^{2} \cos {\left (c + d x \right )}}{d^{5}} + \frac {720 b^{2} x \sin {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \cos {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (a^{2} x + \frac {a b x^{4}}{2} + \frac {b^{2} x^{7}}{7}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*cos(c + d*x)/d - 2*a*b*x**3*cos(c + d*x)/d + 6*a*b*x**2*s in(c + d*x)/d**2 + 12*a*b*x*cos(c + d*x)/d**3 - 12*a*b*sin(c + d*x)/d**4 - b**2*x**6*cos(c + d*x)/d + 6*b**2*x**5*sin(c + d*x)/d**2 + 30*b**2*x**4*c os(c + d*x)/d**3 - 120*b**2*x**3*sin(c + d*x)/d**4 - 360*b**2*x**2*cos(c + d*x)/d**5 + 720*b**2*x*sin(c + d*x)/d**6 + 720*b**2*cos(c + d*x)/d**7, Ne (d, 0)), ((a**2*x + a*b*x**4/2 + b**2*x**7/7)*sin(c), True))
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (188) = 376\).
Time = 0.23 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.60 \[ \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right ) + \frac {b^{2} c^{6} \cos \left (d x + c\right )}{d^{6}} - \frac {2 \, a b c^{3} \cos \left (d x + c\right )}{d^{3}} - \frac {6 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{5}}{d^{6}} + \frac {6 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b c^{2}}{d^{3}} + \frac {15 \, {\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^{2} c^{4}}{d^{6}} - \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 )} a b c}{d^{3}} - \frac {20 \, {\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^{2} c^{3}}{d^{6}} + \frac {2 \, {\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 )} a b}{d^{3}} + \frac {15 \, {\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^{2} c^{2}}{d^{6}} - \frac {6 \, {\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^{2} c}{d^{6}} + \frac {{\left ({\left ({\left (d x + c\right )}^{6} - 30 \, {\left (d x + c\right )}^{4} + 360 \, {\left (d x + c\right )}^{2} - 720\right )} \cos \left (d x + c\right ) - 6 \, {\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \sin \left (d x + c\right )\right )} b^{2}}{d^{6}}}{d} \]
-(a^2*cos(d*x + c) + b^2*c^6*cos(d*x + c)/d^6 - 2*a*b*c^3*cos(d*x + c)/d^3 - 6*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b^2*c^5/d^6 + 6*((d*x + c)*co s(d*x + c) - sin(d*x + c))*a*b*c^2/d^3 + 15*(((d*x + c)^2 - 2)*cos(d*x + c ) - 2*(d*x + c)*sin(d*x + c))*b^2*c^4/d^6 - 6*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a*b*c/d^3 - 20*(((d*x + c)^3 - 6*d*x - 6*c )*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*b^2*c^3/d^6 + 2*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*a*b /d^3 + 15*(((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^2*c^2/d^6 - 6*(((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 + 2 4)*sin(d*x + c))*b^2*c/d^6 + (((d*x + c)^6 - 30*(d*x + c)^4 + 360*(d*x + c )^2 - 720)*cos(d*x + c) - 6*((d*x + c)^5 - 20*(d*x + c)^3 + 120*d*x + 120* c)*sin(d*x + c))*b^2/d^6)/d
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.70 \[ \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx=-\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} - 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{7}} + \frac {6 \, {\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \]
-(b^2*d^6*x^6 + 2*a*b*d^6*x^3 - 30*b^2*d^4*x^4 + a^2*d^6 - 12*a*b*d^4*x + 360*b^2*d^2*x^2 - 720*b^2)*cos(d*x + c)/d^7 + 6*(b^2*d^5*x^5 + a*b*d^5*x^2 - 20*b^2*d^3*x^3 - 2*a*b*d^3 + 120*b^2*d*x)*sin(d*x + c)/d^7
Time = 0.77 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx=\frac {\cos \left (c+d\,x\right )\,\left (720\,b^2-a^2\,d^6\right )}{d^7}-\frac {b^2\,x^6\,\cos \left (c+d\,x\right )}{d}+\frac {30\,b^2\,x^4\,\cos \left (c+d\,x\right )}{d^3}-\frac {360\,b^2\,x^2\,\cos \left (c+d\,x\right )}{d^5}+\frac {6\,b^2\,x^5\,\sin \left (c+d\,x\right )}{d^2}-\frac {120\,b^2\,x^3\,\sin \left (c+d\,x\right )}{d^4}-\frac {12\,a\,b\,\sin \left (c+d\,x\right )}{d^4}+\frac {720\,b^2\,x\,\sin \left (c+d\,x\right )}{d^6}-\frac {2\,a\,b\,x^3\,\cos \left (c+d\,x\right )}{d}+\frac {6\,a\,b\,x^2\,\sin \left (c+d\,x\right )}{d^2}+\frac {12\,a\,b\,x\,\cos \left (c+d\,x\right )}{d^3} \]
(cos(c + d*x)*(720*b^2 - a^2*d^6))/d^7 - (b^2*x^6*cos(c + d*x))/d + (30*b^ 2*x^4*cos(c + d*x))/d^3 - (360*b^2*x^2*cos(c + d*x))/d^5 + (6*b^2*x^5*sin( c + d*x))/d^2 - (120*b^2*x^3*sin(c + d*x))/d^4 - (12*a*b*sin(c + d*x))/d^4 + (720*b^2*x*sin(c + d*x))/d^6 - (2*a*b*x^3*cos(c + d*x))/d + (6*a*b*x^2* sin(c + d*x))/d^2 + (12*a*b*x*cos(c + d*x))/d^3