\(\int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx\) [569]

Optimal result

Integrand size = 17, antiderivative size = 185 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {44}{125} e^{x/2} \cos (x)+\frac {6}{25} e^{x/2} x \cos (x)+\frac {1}{10} e^{x/2} x^2 \cos (x)+\frac {428 e^{x/2} \cos (3 x)}{50653}-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{125} e^{x/2} \sin (x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {1}{5} e^{x/2} x^2 \sin (x)+\frac {792 e^{x/2} \sin (3 x)}{50653}+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {3}{37} e^{x/2} x^2 \sin (3 x) \] Output:

-44/125*exp(1/2*x)*cos(x)+6/25*exp(1/2*x)*x*cos(x)+1/10*exp(1/2*x)*x^2*cos 
(x)+428/50653*exp(1/2*x)*cos(3*x)-70/1369*exp(1/2*x)*x*cos(3*x)-1/74*exp(1 
/2*x)*x^2*cos(3*x)-8/125*exp(1/2*x)*sin(x)-8/25*exp(1/2*x)*x*sin(x)+1/5*ex 
p(1/2*x)*x^2*sin(x)+792/50653*exp(1/2*x)*sin(3*x)+24/1369*exp(1/2*x)*x*sin 
(3*x)-3/37*exp(1/2*x)*x^2*sin(3*x)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.41 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=\frac {e^{x/2} \left (50653 \left (\left (-88+60 x+25 x^2\right ) \cos (x)+2 \left (-8-40 x+25 x^2\right ) \sin (x)\right )-125 \left (\left (-856+5180 x+1369 x^2\right ) \cos (3 x)+6 \left (-264-296 x+1369 x^2\right ) \sin (3 x)\right )\right )}{12663250} \] Input:

Integrate[E^(x/2)*x^2*Cos[x]*Sin[x]^2,x]
 

Output:

(E^(x/2)*(50653*((-88 + 60*x + 25*x^2)*Cos[x] + 2*(-8 - 40*x + 25*x^2)*Sin 
[x]) - 125*((-856 + 5180*x + 1369*x^2)*Cos[3*x] + 6*(-264 - 296*x + 1369*x 
^2)*Sin[3*x])))/12663250
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 185, 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 = {4973, 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.

\(\displaystyle \int e^{x/2} x^2 \sin ^2(x) \cos (x) \, dx\)

\(\Big \downarrow \) 4973

\(\displaystyle \int \left (\frac {1}{4} e^{x/2} x^2 \cos (x)-\frac {1}{4} e^{x/2} x^2 \cos (3 x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} e^{x/2} x^2 \sin (x)-\frac {3}{37} e^{x/2} x^2 \sin (3 x)+\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {8}{125} e^{x/2} \sin (x)+\frac {792 e^{x/2} \sin (3 x)}{50653}+\frac {6}{25} e^{x/2} x \cos (x)-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {44}{125} e^{x/2} \cos (x)+\frac {428 e^{x/2} \cos (3 x)}{50653}\)

Input:

Int[E^(x/2)*x^2*Cos[x]*Sin[x]^2,x]
 

Output:

(-44*E^(x/2)*Cos[x])/125 + (6*E^(x/2)*x*Cos[x])/25 + (E^(x/2)*x^2*Cos[x])/ 
10 + (428*E^(x/2)*Cos[3*x])/50653 - (70*E^(x/2)*x*Cos[3*x])/1369 - (E^(x/2 
)*x^2*Cos[3*x])/74 - (8*E^(x/2)*Sin[x])/125 - (8*E^(x/2)*x*Sin[x])/25 + (E 
^(x/2)*x^2*Sin[x])/5 + (792*E^(x/2)*Sin[3*x])/50653 + (24*E^(x/2)*x*Sin[3* 
x])/1369 - (3*E^(x/2)*x^2*Sin[3*x])/37
 

Defintions of rubi rules used

rule 2009

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

rule 4973

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(x_)^(p 
_.)*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[x^p*F^ 
(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, 
d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.42

method	result	size
default	\(\frac {\left (\frac {2}{5} x^{2}+\frac {24}{25} x -\frac {176}{125}\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (x \right )}{4}-\frac {\left (-\frac {4}{5} x^{2}+\frac {32}{25} x +\frac {32}{125}\right ) {\mathrm e}^{\frac {x}{2}} \sin \left (x \right )}{4}-\frac {\left (\frac {2}{37} x^{2}+\frac {280}{1369} x -\frac {1712}{50653}\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (3 x \right )}{4}+\frac {\left (-\frac {12}{37} x^{2}+\frac {96}{1369} x +\frac {3168}{50653}\right ) {\mathrm e}^{\frac {x}{2}} \sin \left (3 x \right )}{4}\)	\(78\)
risch	\(\left (-\frac {1}{202612}+\frac {3 i}{101306}\right ) \left (1369 x^{2}+888 i x -148 x -96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}+3 i\right ) x}+\left (\frac {1}{500}-\frac {i}{250}\right ) \left (25 x^{2}+40 i x -20 x -32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}+i\right ) x}+\left (\frac {1}{500}+\frac {i}{250}\right ) \left (25 x^{2}-40 i x -20 x +32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}-i\right ) x}+\left (-\frac {1}{202612}-\frac {3 i}{101306}\right ) \left (1369 x^{2}-888 i x -148 x +96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}-3 i\right ) x}\)	\(106\)
orering	\(\frac {8 \left (718725 x^{5}+1843340 x^{4}-2890512 x^{3}+213808 x^{2}-4008768 x -558336\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (x \right ) \sin \left (x \right )^{2}}{6331625 x^{3}}-\frac {8 \left (787175 x^{4}-608280 x^{3}+405864 x^{2}-2921152 x -418752\right ) \left (\frac {{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right ) \sin \left (x \right )^{2}}{2}+2 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right ) \sin \left (x \right )^{2}-{\mathrm e}^{\frac {x}{2}} x^{2} \sin \left (x \right )^{3}+2 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{2} \sin \left (x \right )\right )}{6331625 x^{4}}+\frac {32 \left (34225 x^{3}+74740 x^{2}-229192 x -34896\right ) \left (-\frac {27 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right ) \sin \left (x \right )^{2}}{4}+2 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right ) \sin \left (x \right )^{2}-{\mathrm e}^{\frac {x}{2}} x^{2} \sin \left (x \right )^{3}+2 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{2} \sin \left (x \right )+2 \,{\mathrm e}^{\frac {x}{2}} \cos \left (x \right ) \sin \left (x \right )^{2}-4 \,{\mathrm e}^{\frac {x}{2}} x \sin \left (x \right )^{3}+8 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{2} \sin \left (x \right )+2 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{3}\right )}{6331625 x^{3}}-\frac {16 \left (34225 x^{2}-62160 x -11632\right ) \left (-\frac {81 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right ) \sin \left (x \right )^{2}}{2}+12 \,{\mathrm e}^{\frac {x}{2}} \cos \left (x \right )^{2} \sin \left (x \right )+\frac {25 \,{\mathrm e}^{\frac {x}{2}} x^{2} \sin \left (x \right )^{3}}{4}-\frac {37 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{2} \sin \left (x \right )}{2}+3 \,{\mathrm e}^{\frac {x}{2}} \cos \left (x \right ) \sin \left (x \right )^{2}+12 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{3}+3 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{3}-\frac {83 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right ) \sin \left (x \right )^{2}}{8}-6 \,{\mathrm e}^{\frac {x}{2}} \sin \left (x \right )^{3}-6 \,{\mathrm e}^{\frac {x}{2}} x \sin \left (x \right )^{3}+12 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{2} \sin \left (x \right )\right )}{6331625 x^{2}}\)	\(404\)

Input:

int(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/4*(2/5*x^2+24/25*x-176/125)*exp(1/2*x)*cos(x)-1/4*(-4/5*x^2+32/25*x+32/1 
25)*exp(1/2*x)*sin(x)-1/4*(2/37*x^2+280/1369*x-1712/50653)*exp(1/2*x)*cos( 
3*x)+1/4*(-12/37*x^2+96/1369*x+3168/50653)*exp(1/2*x)*sin(3*x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {4}{6331625} \, {\left (375 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} - 444925 \, x^{2} + 534280 \, x + 126056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) - \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (x\right )^{3} - {\left (444925 \, x^{2} + 1245420 \, x - 1194616\right )} \cos \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \] Input:

integrate(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x, algorithm="fricas")
 

Output:

-4/6331625*(375*(1369*x^2 - 296*x - 264)*cos(x)^2 - 444925*x^2 + 534280*x 
+ 126056)*e^(1/2*x)*sin(x) - 2/6331625*(125*(1369*x^2 + 5180*x - 856)*cos( 
x)^3 - (444925*x^2 + 1245420*x - 1194616)*cos(x))*e^(1/2*x)
 

Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.09 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=\frac {52 x^{2} e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac {26 x^{2} e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{185} - \frac {8 x^{2} e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac {16 x^{2} e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac {11552 x e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac {13464 x e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{34225} - \frac {9152 x e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac {6464 x e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac {504224 e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac {2389232 e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{6331625} - \frac {108224 e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac {2175232 e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{6331625} \] Input:

integrate(exp(1/2*x)*x**2*cos(x)*sin(x)**2,x)
                                                                                    
                                                                                    
 

Output:

52*x**2*exp(x/2)*sin(x)**3/185 + 26*x**2*exp(x/2)*sin(x)**2*cos(x)/185 - 8 
*x**2*exp(x/2)*sin(x)*cos(x)**2/185 + 16*x**2*exp(x/2)*cos(x)**3/185 - 115 
52*x*exp(x/2)*sin(x)**3/34225 + 13464*x*exp(x/2)*sin(x)**2*cos(x)/34225 - 
9152*x*exp(x/2)*sin(x)*cos(x)**2/34225 + 6464*x*exp(x/2)*cos(x)**3/34225 - 
 504224*exp(x/2)*sin(x)**3/6331625 - 2389232*exp(x/2)*sin(x)**2*cos(x)/633 
1625 - 108224*exp(x/2)*sin(x)*cos(x)**2/6331625 - 2175232*exp(x/2)*cos(x)* 
*3/6331625
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.42 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) e^{\left (\frac {1}{2} \, x\right )} - \frac {3}{50653} \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac {1}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) \] Input:

integrate(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x, algorithm="maxima")
 

Output:

-1/101306*(1369*x^2 + 5180*x - 856)*cos(3*x)*e^(1/2*x) + 1/250*(25*x^2 + 6 
0*x - 88)*cos(x)*e^(1/2*x) - 3/50653*(1369*x^2 - 296*x - 264)*e^(1/2*x)*si 
n(3*x) + 1/125*(25*x^2 - 40*x - 8)*e^(1/2*x)*sin(x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {1}{101306} \, {\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \] Input:

integrate(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x, algorithm="giac")
 

Output:

-1/101306*((1369*x^2 + 5180*x - 856)*cos(3*x) + 6*(1369*x^2 - 296*x - 264) 
*sin(3*x))*e^(1/2*x) + 1/250*((25*x^2 + 60*x - 88)*cos(x) + 2*(25*x^2 - 40 
*x - 8)*sin(x))*e^(1/2*x)
 

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.45 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=\frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )-4457464\,\cos \left (x\right )-810448\,\sin \left (x\right )-647500\,x\,\cos \left (3\,x\right )+1266325\,x^2\,\cos \left (x\right )+222000\,x\,\sin \left (3\,x\right )+2532650\,x^2\,\sin \left (x\right )-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )+3039180\,x\,\cos \left (x\right )-4052240\,x\,\sin \left (x\right )\right )}{12663250} \] Input:

int(x^2*exp(x/2)*cos(x)*sin(x)^2,x)
 

Output:

(exp(x/2)*(107000*cos(3*x) + 198000*sin(3*x) - 4457464*cos(x) - 810448*sin 
(x) - 647500*x*cos(3*x) + 1266325*x^2*cos(x) + 222000*x*sin(3*x) + 2532650 
*x^2*sin(x) - 171125*x^2*cos(3*x) - 1026750*x^2*sin(3*x) + 3039180*x*cos(x 
) - 4052240*x*sin(x)))/12663250
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.49 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=\frac {2 e^{\frac {x}{2}} \left (171125 \cos \left (x \right ) \sin \left (x \right )^{2} x^{2}+647500 \cos \left (x \right ) \sin \left (x \right )^{2} x -107000 \cos \left (x \right ) \sin \left (x \right )^{2}+273800 \cos \left (x \right ) x^{2}+597920 \cos \left (x \right ) x -1087616 \cos \left (x \right )+1026750 \sin \left (x \right )^{3} x^{2}-222000 \sin \left (x \right )^{3} x -198000 \sin \left (x \right )^{3}-136900 \sin \left (x \right ) x^{2}-846560 \sin \left (x \right ) x -54112 \sin \left (x \right )\right )}{6331625} \] Input:

int(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x)
 

Output:

(2*e**(x/2)*(171125*cos(x)*sin(x)**2*x**2 + 647500*cos(x)*sin(x)**2*x - 10 
7000*cos(x)*sin(x)**2 + 273800*cos(x)*x**2 + 597920*cos(x)*x - 1087616*cos 
(x) + 1026750*sin(x)**3*x**2 - 222000*sin(x)**3*x - 198000*sin(x)**3 - 136 
900*sin(x)*x**2 - 846560*sin(x)*x - 54112*sin(x)))/6331625