Integrand size = 17, antiderivative size = 185 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {44}{125} e^{x/2} \cos (x)+\frac {6}{25} e^{x/2} x \cos (x)+\frac {1}{10} e^{x/2} x^2 \cos (x)+\frac {428 e^{x/2} \cos (3 x)}{50653}-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{125} e^{x/2} \sin (x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {1}{5} e^{x/2} x^2 \sin (x)+\frac {792 e^{x/2} \sin (3 x)}{50653}+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {3}{37} e^{x/2} x^2 \sin (3 x) \]
-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)
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.41 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=\frac {e^{x/2} \left (50653 \left (\left (-88+60 x+25 x^2\right ) \cos (x)+2 \left (-8-40 x+25 x^2\right ) \sin (x)\right )-125 \left (\left (-856+5180 x+1369 x^2\right ) \cos (3 x)+6 \left (-264-296 x+1369 x^2\right ) \sin (3 x)\right )\right )}{12663250} \]
(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
Time = 0.48 (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 |
\(\displaystyle \frac {1}{5} e^{x/2} x^2 \sin (x)-\frac {3}{37} e^{x/2} x^2 \sin (3 x)+\frac {1}{10} e^{x/2} x^2 \cos (x)-\frac {1}{74} e^{x/2} x^2 \cos (3 x)-\frac {8}{25} e^{x/2} x \sin (x)+\frac {24 e^{x/2} x \sin (3 x)}{1369}-\frac {8}{125} e^{x/2} \sin (x)+\frac {792 e^{x/2} \sin (3 x)}{50653}+\frac {6}{25} e^{x/2} x \cos (x)-\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {44}{125} e^{x/2} \cos (x)+\frac {428 e^{x/2} \cos (3 x)}{50653}\) |
(-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
3.6.69.3.1 Defintions of rubi rules used
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]
Time = 0.86 (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 | \(\left (-\frac {1}{202612}+\frac {3 i}{101306}\right ) \left (1369 x^{2}+888 i x -148 x -96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}+3 i\right ) x}+\left (\frac {1}{500}-\frac {i}{250}\right ) \left (25 x^{2}+40 i x -20 x -32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}+i\right ) x}+\left (\frac {1}{500}+\frac {i}{250}\right ) \left (25 x^{2}-40 i x -20 x +32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}-i\right ) x}+\left (-\frac {1}{202612}-\frac {3 i}{101306}\right ) \left (1369 x^{2}-888 i x -148 x +96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}-3 i\right ) x}\) | \(106\) |
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)
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {4}{6331625} \, {\left (375 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} - 444925 \, x^{2} + 534280 \, x + 126056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) - \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (x\right )^{3} - {\left (444925 \, x^{2} + 1245420 \, x - 1194616\right )} \cos \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \]
-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)
Time = 1.02 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.09 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=\frac {52 x^{2} e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac {26 x^{2} e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{185} - \frac {8 x^{2} e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac {16 x^{2} e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac {11552 x e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac {13464 x e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{34225} - \frac {9152 x e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac {6464 x e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac {504224 e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac {2389232 e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{6331625} - \frac {108224 e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac {2175232 e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{6331625} \]
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
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.42 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) e^{\left (\frac {1}{2} \, x\right )} - \frac {3}{50653} \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac {1}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) \]
-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)
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=-\frac {1}{101306} \, {\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \]
-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)
Time = 0.64 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.45 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx=\frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )-4457464\,\cos \left (x\right )-810448\,\sin \left (x\right )-647500\,x\,\cos \left (3\,x\right )+1266325\,x^2\,\cos \left (x\right )+222000\,x\,\sin \left (3\,x\right )+2532650\,x^2\,\sin \left (x\right )-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )+3039180\,x\,\cos \left (x\right )-4052240\,x\,\sin \left (x\right )\right )}{12663250} \]