Integrand size = 15, antiderivative size = 126 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=-\frac {24 b \cos (c+d x)}{d^5}+\frac {6 a x \cos (c+d x)}{d^3}+\frac {12 b x^2 \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^4 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}-\frac {24 b x \sin (c+d x)}{d^4}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {4 b x^3 \sin (c+d x)}{d^2} \] Output:
-24*b*cos(d*x+c)/d^5+6*a*x*cos(d*x+c)/d^3+12*b*x^2*cos(d*x+c)/d^3-a*x^3*co s(d*x+c)/d-b*x^4*cos(d*x+c)/d-6*a*sin(d*x+c)/d^4-24*b*x*sin(d*x+c)/d^4+3*a *x^2*sin(d*x+c)/d^2+4*b*x^3*sin(d*x+c)/d^2
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.65 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=\frac {-\left (\left (a d^2 x \left (-6+d^2 x^2\right )+b \left (24-12 d^2 x^2+d^4 x^4\right )\right ) \cos (c+d x)\right )+d \left (4 b x \left (-6+d^2 x^2\right )+3 a \left (-2+d^2 x^2\right )\right ) \sin (c+d x)}{d^5} \] Input:
Integrate[x^3*(a + b*x)*Sin[c + d*x],x]
Output:
(-((a*d^2*x*(-6 + d^2*x^2) + b*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + d*(4*b*x*(-6 + d^2*x^2) + 3*a*(-2 + d^2*x^2))*Sin[c + d*x])/d^5
Time = 0.49 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.
| \(\displaystyle \int x^3 (a+b x) \sin (c+d x) \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (a x^3 \sin (c+d x)+b x^4 \sin (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 a \sin (c+d x)}{d^4}+\frac {6 a x \cos (c+d x)}{d^3}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {a x^3 \cos (c+d x)}{d}-\frac {24 b \cos (c+d x)}{d^5}-\frac {24 b x \sin (c+d x)}{d^4}+\frac {12 b x^2 \cos (c+d x)}{d^3}+\frac {4 b x^3 \sin (c+d x)}{d^2}-\frac {b x^4 \cos (c+d x)}{d}\) |
Input:
Int[x^3*(a + b*x)*Sin[c + d*x],x]
Output:
(-24*b*Cos[c + d*x])/d^5 + (6*a*x*Cos[c + d*x])/d^3 + (12*b*x^2*Cos[c + d* x])/d^3 - (a*x^3*Cos[c + d*x])/d - (b*x^4*Cos[c + d*x])/d - (6*a*Sin[c + d *x])/d^4 - (24*b*x*Sin[c + d*x])/d^4 + (3*a*x^2*Sin[c + d*x])/d^2 + (4*b*x ^3*Sin[c + d*x])/d^2
Time = 0.94 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{3}-12 x^{2} d^{2} b -6 a \,d^{2} x +24 b \right ) \cos \left (d x +c \right )}{d^{5}}+\frac {\left (4 b \,d^{2} x^{3}+3 a \,d^{2} x^{2}-24 b x -6 a \right ) \sin \left (d x +c \right )}{d^{4}}\) | \(85\) |
| parallelrisch | \(\frac {d^{2} x \left (x^{2} \left (b x +a \right ) d^{2}-12 b x -6 a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 d \left (x^{2} \left (\frac {4 b x}{3}+a \right ) d^{2}-8 b x -2 a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-b \,x^{4}-a \,x^{3}\right ) d^{4}+6 x \left (2 b x +a \right ) d^{2}-48 b}{d^{5} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\) | \(122\) |
| orering | \(\frac {2 \left (4 b^{2} d^{4} x^{5}+7 a b \,d^{4} x^{4}+3 a^{2} d^{4} x^{3}-36 b^{2} d^{2} x^{3}-45 a b \,d^{2} x^{2}-12 x \,a^{2} d^{2}+48 b^{2} x +36 a b \right ) \sin \left (d x +c \right )}{d^{6} x \left (b x +a \right )}-\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{3}-12 x^{2} d^{2} b -6 a \,d^{2} x +24 b \right ) \left (3 x^{2} \left (b x +a \right ) \sin \left (d x +c \right )+x^{3} b \sin \left (d x +c \right )+x^{3} \left (b x +a \right ) d \cos \left (d x +c \right )\right )}{d^{6} x^{3} \left (b x +a \right )}\) | \(191\) |
| norman | \(\frac {\frac {a \,x^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {b \,x^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {48 b}{d^{5}}-\frac {12 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}+\frac {6 a x}{d^{3}}-\frac {a \,x^{3}}{d}+\frac {12 b \,x^{2}}{d^{3}}-\frac {b \,x^{4}}{d}-\frac {6 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d^{3}}+\frac {6 a \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}-\frac {48 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}-\frac {12 b \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d^{3}}+\frac {8 b \,x^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\) | \(201\) |
| meijerg | \(\frac {16 b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {x \left (d^{2}\right )^{\frac {5}{2}} \left (-\frac {5 x^{2} d^{2}}{2}+15\right ) \cos \left (d x \right )}{10 \sqrt {\pi }\, d^{4}}+\frac {\left (d^{2}\right )^{\frac {5}{2}} \left (\frac {5}{8} x^{4} d^{4}-\frac {15}{2} x^{2} d^{2}+15\right ) \sin \left (d x \right )}{10 \sqrt {\pi }\, d^{5}}\right )}{d^{4} \sqrt {d^{2}}}+\frac {16 b \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{4} d^{4}-\frac {9}{2} x^{2} d^{2}+9\right ) \cos \left (d x \right )}{6 \sqrt {\pi }}-\frac {x d \left (-\frac {3 x^{2} d^{2}}{2}+9\right ) \sin \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {8 a \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 x^{2} d^{2}}{2}+3\right ) \cos \left (d x \right )}{4 \sqrt {\pi }}-\frac {x d \left (-\frac {x^{2} d^{2}}{2}+3\right ) \sin \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}+\frac {8 a \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {x d \left (-\frac {5 x^{2} d^{2}}{2}+15\right ) \cos \left (d x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 x^{2} d^{2}}{2}+15\right ) \sin \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}\) | \(256\) |
| parts | \(-\frac {b \,x^{4} \cos \left (d x +c \right )}{d}-\frac {a \,x^{3} \cos \left (d x +c \right )}{d}+\frac {\frac {3 a \,c^{2} \sin \left (d x +c \right )}{d^{2}}-\frac {6 a c \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}+\frac {3 a \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}-\frac {4 b \,c^{3} \sin \left (d x +c \right )}{d^{3}}+\frac {12 b \,c^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {12 b c \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {4 b \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}}{d^{2}}\) | \(263\) |
| derivativedivides | \(\frac {a \,c^{3} \cos \left (d x +c \right )+3 a \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )-3 a c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )-\frac {b \,c^{4} \cos \left (d x +c \right )}{d}-\frac {4 b \,c^{3} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}+\frac {6 b \,c^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}-\frac {4 b c \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}+\frac {b \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}}{d^{4}}\) | \(359\) |
| default | \(\frac {a \,c^{3} \cos \left (d x +c \right )+3 a \,c^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )-3 a c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )-\frac {b \,c^{4} \cos \left (d x +c \right )}{d}-\frac {4 b \,c^{3} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}+\frac {6 b \,c^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}-\frac {4 b c \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d}+\frac {b \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}}{d^{4}}\) | \(359\) |
Input:
int(x^3*(b*x+a)*sin(d*x+c),x,method=_RETURNVERBOSE)
Output:
-(b*d^4*x^4+a*d^4*x^3-12*b*d^2*x^2-6*a*d^2*x+24*b)/d^5*cos(d*x+c)+1/d^4*(4 *b*d^2*x^3+3*a*d^2*x^2-24*b*x-6*a)*sin(d*x+c)
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.67 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=-\frac {{\left (b d^{4} x^{4} + a d^{4} x^{3} - 12 \, b d^{2} x^{2} - 6 \, a d^{2} x + 24 \, b\right )} \cos \left (d x + c\right ) - {\left (4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} - 24 \, b d x - 6 \, a d\right )} \sin \left (d x + c\right )}{d^{5}} \] Input:
integrate(x^3*(b*x+a)*sin(d*x+c),x, algorithm="fricas")
Output:
-((b*d^4*x^4 + a*d^4*x^3 - 12*b*d^2*x^2 - 6*a*d^2*x + 24*b)*cos(d*x + c) - (4*b*d^3*x^3 + 3*a*d^3*x^2 - 24*b*d*x - 6*a*d)*sin(d*x + c))/d^5
Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.20 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=\begin {cases} - \frac {a x^{3} \cos {\left (c + d x \right )}}{d} + \frac {3 a x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \cos {\left (c + d x \right )}}{d^{3}} - \frac {6 a \sin {\left (c + d x \right )}}{d^{4}} - \frac {b x^{4} \cos {\left (c + d x \right )}}{d} + \frac {4 b x^{3} \sin {\left (c + d x \right )}}{d^{2}} + \frac {12 b x^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {24 b x \sin {\left (c + d x \right )}}{d^{4}} - \frac {24 b \cos {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{5}}{5}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(b*x+a)*sin(d*x+c),x)
Output:
Piecewise((-a*x**3*cos(c + d*x)/d + 3*a*x**2*sin(c + d*x)/d**2 + 6*a*x*cos (c + d*x)/d**3 - 6*a*sin(c + d*x)/d**4 - b*x**4*cos(c + d*x)/d + 4*b*x**3* sin(c + d*x)/d**2 + 12*b*x**2*cos(c + d*x)/d**3 - 24*b*x*sin(c + d*x)/d**4 - 24*b*cos(c + d*x)/d**5, Ne(d, 0)), ((a*x**4/4 + b*x**5/5)*sin(c), True) )
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (126) = 252\).
Time = 0.05 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.43 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=\frac {a c^{3} \cos \left (d x + c\right ) - \frac {b c^{4} \cos \left (d x + c\right )}{d} - 3 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c^{2} + \frac {4 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{3}}{d} + 3 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a c - \frac {6 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c^{2}}{d} - {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} a + \frac {4 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b c}{d} - \frac {{\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b}{d}}{d^{4}} \] Input:
integrate(x^3*(b*x+a)*sin(d*x+c),x, algorithm="maxima")
Output:
(a*c^3*cos(d*x + c) - b*c^4*cos(d*x + c)/d - 3*((d*x + c)*cos(d*x + c) - s in(d*x + c))*a*c^2 + 4*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b*c^3/d + 3 *(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a*c - 6*(((d* x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b*c^2/d - (((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*a + 4 *(((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/d - (((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/d)/d^4
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.68 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=-\frac {{\left (b d^{4} x^{4} + a d^{4} x^{3} - 12 \, b d^{2} x^{2} - 6 \, a d^{2} x + 24 \, b\right )} \cos \left (d x + c\right )}{d^{5}} + \frac {{\left (4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} - 24 \, b d x - 6 \, a d\right )} \sin \left (d x + c\right )}{d^{5}} \] Input:
integrate(x^3*(b*x+a)*sin(d*x+c),x, algorithm="giac")
Output:
-(b*d^4*x^4 + a*d^4*x^3 - 12*b*d^2*x^2 - 6*a*d^2*x + 24*b)*cos(d*x + c)/d^ 5 + (4*b*d^3*x^3 + 3*a*d^3*x^2 - 24*b*d*x - 6*a*d)*sin(d*x + c)/d^5
Time = 0.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.97 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=\frac {6\,a\,x\,\cos \left (c+d\,x\right )+12\,b\,x^2\,\cos \left (c+d\,x\right )}{d^3}-\frac {6\,a\,\sin \left (c+d\,x\right )+24\,b\,x\,\sin \left (c+d\,x\right )}{d^4}-\frac {a\,x^3\,\cos \left (c+d\,x\right )+b\,x^4\,\cos \left (c+d\,x\right )}{d}+\frac {3\,a\,x^2\,\sin \left (c+d\,x\right )+4\,b\,x^3\,\sin \left (c+d\,x\right )}{d^2}-\frac {24\,b\,\cos \left (c+d\,x\right )}{d^5} \] Input:
int(x^3*sin(c + d*x)*(a + b*x),x)
Output:
(6*a*x*cos(c + d*x) + 12*b*x^2*cos(c + d*x))/d^3 - (6*a*sin(c + d*x) + 24* b*x*sin(c + d*x))/d^4 - (a*x^3*cos(c + d*x) + b*x^4*cos(c + d*x))/d + (3*a *x^2*sin(c + d*x) + 4*b*x^3*sin(c + d*x))/d^2 - (24*b*cos(c + d*x))/d^5
Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int x^3 (a+b x) \sin (c+d x) \, dx=\frac {-\cos \left (d x +c \right ) a \,d^{4} x^{3}+6 \cos \left (d x +c \right ) a \,d^{2} x -\cos \left (d x +c \right ) b \,d^{4} x^{4}+12 \cos \left (d x +c \right ) b \,d^{2} x^{2}-24 \cos \left (d x +c \right ) b +3 \sin \left (d x +c \right ) a \,d^{3} x^{2}-6 \sin \left (d x +c \right ) a d +4 \sin \left (d x +c \right ) b \,d^{3} x^{3}-24 \sin \left (d x +c \right ) b d x}{d^{5}} \] Input:
int(x^3*(b*x+a)*sin(d*x+c),x)
Output:
( - cos(c + d*x)*a*d**4*x**3 + 6*cos(c + d*x)*a*d**2*x - cos(c + d*x)*b*d* *4*x**4 + 12*cos(c + d*x)*b*d**2*x**2 - 24*cos(c + d*x)*b + 3*sin(c + d*x) *a*d**3*x**2 - 6*sin(c + d*x)*a*d + 4*sin(c + d*x)*b*d**3*x**3 - 24*sin(c + d*x)*b*d*x)/d**5