\(\int x^2 (a+b x^2) \sin (c+d x) \, dx\) [41]

Optimal result

Integrand size = 17, antiderivative size = 111 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=-\frac {24 b \cos (c+d x)}{d^5}+\frac {2 a \cos (c+d x)}{d^3}+\frac {12 b x^2 \cos (c+d x)}{d^3}-\frac {a x^2 \cos (c+d x)}{d}-\frac {b x^4 \cos (c+d x)}{d}-\frac {24 b x \sin (c+d x)}{d^4}+\frac {2 a x \sin (c+d x)}{d^2}+\frac {4 b x^3 \sin (c+d x)}{d^2} \] Output:

-24*b*cos(d*x+c)/d^5+2*a*cos(d*x+c)/d^3+12*b*x^2*cos(d*x+c)/d^3-a*x^2*cos( 
d*x+c)/d-b*x^4*cos(d*x+c)/d-24*b*x*sin(d*x+c)/d^4+2*a*x*sin(d*x+c)/d^2+4*b 
*x^3*sin(d*x+c)/d^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=\frac {-\left (\left (a d^2 \left (-2+d^2 x^2\right )+b \left (24-12 d^2 x^2+d^4 x^4\right )\right ) \cos (c+d x)\right )+2 d x \left (a d^2+2 b \left (-6+d^2 x^2\right )\right ) \sin (c+d x)}{d^5} \] Input:

Integrate[x^2*(a + b*x^2)*Sin[c + d*x],x]
 

Output:

(-((a*d^2*(-2 + d^2*x^2) + b*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + 
2*d*x*(a*d^2 + 2*b*(-6 + d^2*x^2))*Sin[c + d*x])/d^5
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 111, 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.

\(\displaystyle \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx\)

\(\Big \downarrow \) 3820

\(\displaystyle \int \left (a x^2 \sin (c+d x)+b x^4 \sin (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a \cos (c+d x)}{d^3}+\frac {2 a x \sin (c+d x)}{d^2}-\frac {a x^2 \cos (c+d x)}{d}-\frac {24 b \cos (c+d x)}{d^5}-\frac {24 b x \sin (c+d x)}{d^4}+\frac {12 b x^2 \cos (c+d x)}{d^3}+\frac {4 b x^3 \sin (c+d x)}{d^2}-\frac {b x^4 \cos (c+d x)}{d}\)

Input:

Int[x^2*(a + b*x^2)*Sin[c + d*x],x]
 

Output:

(-24*b*Cos[c + d*x])/d^5 + (2*a*Cos[c + d*x])/d^3 + (12*b*x^2*Cos[c + d*x] 
)/d^3 - (a*x^2*Cos[c + d*x])/d - (b*x^4*Cos[c + d*x])/d - (24*b*x*Sin[c + 
d*x])/d^4 + (2*a*x*Sin[c + d*x])/d^2 + (4*b*x^3*Sin[c + d*x])/d^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3820

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]
 

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.70

method	result	size
risch	\(-\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{2}-12 x^{2} d^{2} b -2 d^{2} a +24 b \right ) \cos \left (d x +c \right )}{d^{5}}+\frac {2 x \left (2 x^{2} d^{2} b +d^{2} a -12 b \right ) \sin \left (d x +c \right )}{d^{4}}\)	\(78\)
parallelrisch	\(\frac {d^{2} x^{2} \left (\left (b \,x^{2}+a \right ) d^{2}-12 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 d x \left (\left (2 b \,x^{2}+a \right ) d^{2}-12 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-b \,x^{4}-a \,x^{2}\right ) d^{4}+4 \left (3 b \,x^{2}+a \right ) d^{2}-48 b}{d^{5} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\)	\(116\)
norman	\(\frac {\frac {4 d^{2} a -48 b}{d^{5}}+\frac {b \,x^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {\left (d^{2} a -12 b \right ) x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d^{3}}-\frac {b \,x^{4}}{d}-\frac {\left (d^{2} a -12 b \right ) x^{2}}{d^{3}}+\frac {8 b \,x^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{2}}+\frac {4 \left (d^{2} a -12 b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d^{4}}}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\)	\(146\)
orering	\(\frac {4 \left (2 b^{2} d^{4} x^{6}+3 a b \,d^{4} x^{4}+a^{2} d^{4} x^{2}-18 b^{2} d^{2} x^{4}-14 a b \,d^{2} x^{2}-a^{2} d^{2}+24 x^{2} b^{2}+12 a b \right ) \sin \left (d x +c \right )}{d^{6} x \left (b \,x^{2}+a \right )}-\frac {\left (b \,x^{4} d^{4}+a \,d^{4} x^{2}-12 x^{2} d^{2} b -2 d^{2} a +24 b \right ) \left (2 x \left (b \,x^{2}+a \right ) \sin \left (d x +c \right )+2 x^{3} b \sin \left (d x +c \right )+x^{2} \left (b \,x^{2}+a \right ) d \cos \left (d x +c \right )\right )}{d^{6} x^{2} \left (b \,x^{2}+a \right )}\)	\(197\)
parts	\(-\frac {b \,x^{4} \cos \left (d x +c \right )}{d}-\frac {a \,x^{2} \cos \left (d x +c \right )}{d}+\frac {-\frac {2 a c \sin \left (d x +c \right )}{d}+\frac {2 a \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}-\frac {4 b \,c^{3} \sin \left (d x +c \right )}{d^{3}}+\frac {12 b \,c^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {12 b c \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{3}}+\frac {4 b \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}}{d^{2}}\)	\(218\)
meijerg	\(\frac {16 b \sin \left (c \right ) \sqrt {\pi }\, \left (-\frac {x \left (d^{2}\right )^{\frac {5}{2}} \left (-\frac {5 x^{2} d^{2}}{2}+15\right ) \cos \left (d x \right )}{10 \sqrt {\pi }\, d^{4}}+\frac {\left (d^{2}\right )^{\frac {5}{2}} \left (\frac {5}{8} x^{4} d^{4}-\frac {15}{2} x^{2} d^{2}+15\right ) \sin \left (d x \right )}{10 \sqrt {\pi }\, d^{5}}\right )}{d^{4} \sqrt {d^{2}}}+\frac {16 b \cos \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{4} d^{4}-\frac {9}{2} x^{2} d^{2}+9\right ) \cos \left (d x \right )}{6 \sqrt {\pi }}-\frac {x d \left (-\frac {3 x^{2} d^{2}}{2}+9\right ) \sin \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {4 a \sin \left (c \right ) \sqrt {\pi }\, \left (\frac {x \left (d^{2}\right )^{\frac {3}{2}} \cos \left (d x \right )}{2 \sqrt {\pi }\, d^{2}}-\frac {\left (d^{2}\right )^{\frac {3}{2}} \left (-\frac {3 x^{2} d^{2}}{2}+3\right ) \sin \left (d x \right )}{6 \sqrt {\pi }\, d^{3}}\right )}{d^{2} \sqrt {d^{2}}}+\frac {4 a \cos \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{2} d^{2}}{2}+1\right ) \cos \left (d x \right )}{2 \sqrt {\pi }}+\frac {x d \sin \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}\)	\(256\)
derivativedivides	\(\frac {-a \,c^{2} \cos \left (d x +c \right )-2 a c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )-\frac {b \,c^{4} \cos \left (d x +c \right )}{d^{2}}-\frac {4 b \,c^{3} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\frac {6 b \,c^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}-\frac {4 b c \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}}{d^{3}}\)	\(302\)
default	\(\frac {-a \,c^{2} \cos \left (d x +c \right )-2 a c \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+a \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )-\frac {b \,c^{4} \cos \left (d x +c \right )}{d^{2}}-\frac {4 b \,c^{3} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\frac {6 b \,c^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}-\frac {4 b c \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}}{d^{3}}\)	\(302\)

Input:

int(x^2*(b*x^2+a)*sin(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-(b*d^4*x^4+a*d^4*x^2-12*b*d^2*x^2-2*a*d^2+24*b)/d^5*cos(d*x+c)+2/d^4*x*(2 
*b*d^2*x^2+a*d^2-12*b)*sin(d*x+c)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.69 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=-\frac {{\left (b d^{4} x^{4} - 2 \, a d^{2} + {\left (a d^{4} - 12 \, b d^{2}\right )} x^{2} + 24 \, b\right )} \cos \left (d x + c\right ) - 2 \, {\left (2 \, b d^{3} x^{3} + {\left (a d^{3} - 12 \, b d\right )} x\right )} \sin \left (d x + c\right )}{d^{5}} \] Input:

integrate(x^2*(b*x^2+a)*sin(d*x+c),x, algorithm="fricas")
 

Output:

-((b*d^4*x^4 - 2*a*d^2 + (a*d^4 - 12*b*d^2)*x^2 + 24*b)*cos(d*x + c) - 2*( 
2*b*d^3*x^3 + (a*d^3 - 12*b*d)*x)*sin(d*x + c))/d^5
 

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=\begin {cases} - \frac {a x^{2} \cos {\left (c + d x \right )}}{d} + \frac {2 a x \sin {\left (c + d x \right )}}{d^{2}} + \frac {2 a \cos {\left (c + d x \right )}}{d^{3}} - \frac {b x^{4} \cos {\left (c + d x \right )}}{d} + \frac {4 b x^{3} \sin {\left (c + d x \right )}}{d^{2}} + \frac {12 b x^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {24 b x \sin {\left (c + d x \right )}}{d^{4}} - \frac {24 b \cos {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{3}}{3} + \frac {b x^{5}}{5}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \] Input:

integrate(x**2*(b*x**2+a)*sin(d*x+c),x)
 

Output:

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**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**3/3 + b*x**5/5)*sin(c), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (111) = 222\).

Time = 0.05 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.32 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=-\frac {a c^{2} \cos \left (d x + c\right ) + \frac {b c^{4} \cos \left (d x + c\right )}{d^{2}} - 2 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c - \frac {4 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{3}}{d^{2}} + {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a + \frac {6 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c^{2}}{d^{2}} - \frac {4 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b c}{d^{2}} + \frac {{\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b}{d^{2}}}{d^{3}} \] Input:

integrate(x^2*(b*x^2+a)*sin(d*x+c),x, algorithm="maxima")
 

Output:

-(a*c^2*cos(d*x + c) + b*c^4*cos(d*x + c)/d^2 - 2*((d*x + c)*cos(d*x + c) 
- sin(d*x + c))*a*c - 4*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b*c^3/d^2 
+ (((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a + 6*(((d*x 
 + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b*c^2/d^2 - 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^2 + (((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^2)/d^3
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=-\frac {{\left (b d^{4} x^{4} + a d^{4} x^{2} - 12 \, b d^{2} x^{2} - 2 \, a d^{2} + 24 \, b\right )} \cos \left (d x + c\right )}{d^{5}} + \frac {2 \, {\left (2 \, b d^{3} x^{3} + a d^{3} x - 12 \, b d x\right )} \sin \left (d x + c\right )}{d^{5}} \] Input:

integrate(x^2*(b*x^2+a)*sin(d*x+c),x, algorithm="giac")
 

Output:

-(b*d^4*x^4 + a*d^4*x^2 - 12*b*d^2*x^2 - 2*a*d^2 + 24*b)*cos(d*x + c)/d^5 
+ 2*(2*b*d^3*x^3 + a*d^3*x - 12*b*d*x)*sin(d*x + c)/d^5
 

Mupad [B] (verification not implemented)

Time = 42.61 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=\frac {x^2\,\cos \left (c+d\,x\right )\,\left (12\,b-a\,d^2\right )}{d^3}-\frac {2\,\cos \left (c+d\,x\right )\,\left (12\,b-a\,d^2\right )}{d^5}-\frac {2\,x\,\sin \left (c+d\,x\right )\,\left (12\,b-a\,d^2\right )}{d^4}-\frac {b\,x^4\,\cos \left (c+d\,x\right )}{d}+\frac {4\,b\,x^3\,\sin \left (c+d\,x\right )}{d^2} \] Input:

int(x^2*sin(c + d*x)*(a + b*x^2),x)
 

Output:

(x^2*cos(c + d*x)*(12*b - a*d^2))/d^3 - (2*cos(c + d*x)*(12*b - a*d^2))/d^ 
5 - (2*x*sin(c + d*x)*(12*b - a*d^2))/d^4 - (b*x^4*cos(c + d*x))/d + (4*b* 
x^3*sin(c + d*x))/d^2
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99 \[ \int x^2 \left (a+b x^2\right ) \sin (c+d x) \, dx=\frac {-\cos \left (d x +c \right ) a \,d^{4} x^{2}+2 \cos \left (d x +c \right ) a \,d^{2}-\cos \left (d x +c \right ) b \,d^{4} x^{4}+12 \cos \left (d x +c \right ) b \,d^{2} x^{2}-24 \cos \left (d x +c \right ) b +2 \sin \left (d x +c \right ) a \,d^{3} x +4 \sin \left (d x +c \right ) b \,d^{3} x^{3}-24 \sin \left (d x +c \right ) b d x}{d^{5}} \] Input:

int(x^2*(b*x^2+a)*sin(d*x+c),x)
 

Output:

( - cos(c + d*x)*a*d**4*x**2 + 2*cos(c + d*x)*a*d**2 - 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 + 2*sin(c + d*x)*a 
*d**3*x + 4*sin(c + d*x)*b*d**3*x**3 - 24*sin(c + d*x)*b*d*x)/d**5