Integrand size = 15, antiderivative size = 187 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=-\frac {132}{125} e^{x/2} \cos (x)+\frac {18}{25} e^{x/2} x \cos (x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {428 e^{x/2} \cos (3 x)}{50653}+\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {24}{125} e^{x/2} \sin (x)-\frac {24}{25} e^{x/2} x \sin (x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-\frac {792 e^{x/2} \sin (3 x)}{50653}-\frac {24 e^{x/2} x \sin (3 x)}{1369} \] Output:
-132/125*exp(1/2*x)*cos(x)+18/25*exp(1/2*x)*x*cos(x)+48/185*exp(1/2*x)*x^2 *cos(x)+2/37*exp(1/2*x)*x^2*cos(x)^3-428/50653*exp(1/2*x)*cos(3*x)+70/1369 *exp(1/2*x)*x*cos(3*x)-24/125*exp(1/2*x)*sin(x)-24/25*exp(1/2*x)*x*sin(x)+ 96/185*exp(1/2*x)*x^2*sin(x)+12/37*exp(1/2*x)*x^2*cos(x)^2*sin(x)-792/5065 3*exp(1/2*x)*sin(3*x)-24/1369*exp(1/2*x)*x*sin(3*x)
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {e^{x/2} \left (151959 \left (-88+60 x+25 x^2\right ) \cos (x)+125 \left (-856+5180 x+1369 x^2\right ) \cos (3 x)+303918 \left (-8-40 x+25 x^2\right ) \sin (x)+750 \left (-264-296 x+1369 x^2\right ) \sin (3 x)\right )}{12663250} \] Input:
Integrate[E^(x/2)*x^2*Cos[x]^3,x]
Output:
(E^(x/2)*(151959*(-88 + 60*x + 25*x^2)*Cos[x] + 125*(-856 + 5180*x + 1369* x^2)*Cos[3*x] + 303918*(-8 - 40*x + 25*x^2)*Sin[x] + 750*(-264 - 296*x + 1 369*x^2)*Sin[3*x]))/12663250
Time = 0.65 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.38, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4969, 27, 2010, 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.
| \(\displaystyle \int e^{x/2} x^2 \cos ^3(x) \, dx\) |
| \(\Big \downarrow \) 4969 |
| \(\displaystyle -2 \int \frac {2}{185} x \left (5 e^{x/2} \cos ^3(x)+30 e^{x/2} \sin (x) \cos ^2(x)+24 e^{x/2} \cos (x)+48 e^{x/2} \sin (x)\right )dx+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4}{185} \int x \left (5 e^{x/2} \cos ^3(x)+30 e^{x/2} \sin (x) \cos ^2(x)+24 e^{x/2} \cos (x)+48 e^{x/2} \sin (x)\right )dx+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {4}{185} \int \left (5 e^{x/2} x \cos ^3(x)+30 e^{x/2} x \sin (x) \cos ^2(x)+24 e^{x/2} x \cos (x)+48 e^{x/2} x \sin (x)\right )dx+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)-\frac {4}{185} \left (\frac {304668 e^{x/2} \sin (x)}{34225}+\frac {8139}{185} e^{x/2} x \sin (x)+\frac {1020 e^{x/2} \sin (3 x)}{1369}+\frac {15}{37} e^{x/2} x \sin (3 x)-\frac {20 e^{x/2} \cos ^3(x)}{1369}+\frac {10}{37} e^{x/2} x \cos ^3(x)+\frac {1671924 e^{x/2} \cos (x)}{34225}-\frac {6198}{185} e^{x/2} x \cos (x)+\frac {540 e^{x/2} \cos (3 x)}{1369}-\frac {90}{37} e^{x/2} x \cos (3 x)-\frac {120 e^{x/2} \sin (x) \cos ^2(x)}{1369}+\frac {60}{37} e^{x/2} x \sin (x) \cos ^2(x)\right )\) |
Input:
Int[E^(x/2)*x^2*Cos[x]^3,x]
Output:
(48*E^(x/2)*x^2*Cos[x])/185 + (2*E^(x/2)*x^2*Cos[x]^3)/37 + (96*E^(x/2)*x^ 2*Sin[x])/185 + (12*E^(x/2)*x^2*Cos[x]^2*Sin[x])/37 - (4*((1671924*E^(x/2) *Cos[x])/34225 - (6198*E^(x/2)*x*Cos[x])/185 - (20*E^(x/2)*Cos[x]^3)/1369 + (10*E^(x/2)*x*Cos[x]^3)/37 + (540*E^(x/2)*Cos[3*x])/1369 - (90*E^(x/2)*x *Cos[3*x])/37 + (304668*E^(x/2)*Sin[x])/34225 + (8139*E^(x/2)*x*Sin[x])/18 5 - (120*E^(x/2)*Cos[x]^2*Sin[x])/1369 + (60*E^(x/2)*x*Cos[x]^2*Sin[x])/37 + (1020*E^(x/2)*Sin[3*x])/1369 + (15*E^(x/2)*x*Sin[3*x])/37))/185
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Int[Cos[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)* (x_))^(m_.), x_Symbol] :> Module[{u = IntHide[F^(c*(a + b*x))*Cos[d + e*x]^ n, x]}, Simp[(f*x)^m u, x] - Simp[f*m Int[(f*x)^(m - 1)*u, x], x]] /; F reeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
Time = 0.88 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.32
| method | result | size |
| parallelrisch | \(\frac {3 \left (\frac {5 \left (x^{2}+\frac {140}{37} x -\frac {856}{1369}\right ) \cos \left (3 x \right )}{111}+\frac {10 \left (x^{2}-\frac {8}{37} x -\frac {264}{1369}\right ) \sin \left (3 x \right )}{37}+\left (x^{2}+\frac {12}{5} x -\frac {88}{25}\right ) \cos \left (x \right )+2 \sin \left (x \right ) \left (x^{2}-\frac {8}{5} x -\frac {8}{25}\right )\right ) {\mathrm e}^{\frac {x}{2}}}{10}\) | \(59\) |
| default | \(\frac {\left (\frac {2}{37} x^{2}+\frac {280}{1369} x -\frac {1712}{50653}\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (3 x \right )}{4}-\frac {\left (-\frac {12}{37} x^{2}+\frac {96}{1369} x +\frac {3168}{50653}\right ) {\mathrm e}^{\frac {x}{2}} \sin \left (3 x \right )}{4}+\frac {3 \left (\frac {2}{5} x^{2}+\frac {24}{25} x -\frac {176}{125}\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (x \right )}{4}-\frac {3 \left (-\frac {4}{5} x^{2}+\frac {32}{25} x +\frac {32}{125}\right ) {\mathrm e}^{\frac {x}{2}} \sin \left (x \right )}{4}\) | \(78\) |
| risch | \(\left (\frac {1}{202612}-\frac {3 i}{101306}\right ) \left (1369 x^{2}+888 i x -148 x -96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}+3 i\right ) x}+\left (\frac {3}{500}-\frac {3 i}{250}\right ) \left (25 x^{2}+40 i x -20 x -32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}+i\right ) x}+\left (\frac {3}{500}+\frac {3 i}{250}\right ) \left (25 x^{2}-40 i x -20 x +32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}-i\right ) x}+\left (\frac {1}{202612}+\frac {3 i}{101306}\right ) \left (1369 x^{2}-888 i x -148 x +96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}-3 i\right ) x}\) | \(106\) |
| orering | \(\frac {8 \left (718725 x^{5}+1843340 x^{4}-2890512 x^{3}+213808 x^{2}-4008768 x -558336\right ) {\mathrm e}^{\frac {x}{2}} \cos \left (x \right )^{3}}{6331625 x^{3}}-\frac {8 \left (787175 x^{4}-608280 x^{3}+405864 x^{2}-2921152 x -418752\right ) \left (\frac {{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{3}}{2}+2 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{3}-3 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{2} \sin \left (x \right )\right )}{6331625 x^{4}}+\frac {32 \left (34225 x^{3}+74740 x^{2}-229192 x -34896\right ) \left (-\frac {11 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{3}}{4}+2 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{3}-3 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{2} \sin \left (x \right )+2 \,{\mathrm e}^{\frac {x}{2}} \cos \left (x \right )^{3}-12 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{2} \sin \left (x \right )+6 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right ) \sin \left (x \right )^{2}\right )}{6331625 x^{3}}-\frac {16 \left (34225 x^{2}-62160 x -11632\right ) \left (-\frac {35 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{3}}{8}-\frac {33 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{3}}{2}+\frac {75 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right )^{2} \sin \left (x \right )}{4}+3 \,{\mathrm e}^{\frac {x}{2}} \cos \left (x \right )^{3}-18 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right )^{2} \sin \left (x \right )+9 \,{\mathrm e}^{\frac {x}{2}} x^{2} \cos \left (x \right ) \sin \left (x \right )^{2}-18 \,{\mathrm e}^{\frac {x}{2}} \cos \left (x \right )^{2} \sin \left (x \right )+36 \,{\mathrm e}^{\frac {x}{2}} x \cos \left (x \right ) \sin \left (x \right )^{2}-6 \,{\mathrm e}^{\frac {x}{2}} x^{2} \sin \left (x \right )^{3}\right )}{6331625 x^{2}}\) | \(334\) |
Input:
int(exp(1/2*x)*x^2*cos(x)^3,x,method=_RETURNVERBOSE)
Output:
3/10*(5/111*(x^2+140/37*x-856/1369)*cos(3*x)+10/37*(x^2-8/37*x-264/1369)*s in(3*x)+(x^2+12/5*x-88/25)*cos(x)+2*sin(x)*(x^2-8/5*x-8/25))*exp(1/2*x)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {12}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} + 273800 \, x^{2} - 497280 \, x - 93056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) + \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (x\right )^{3} + 24 \, {\left (34225 \, x^{2} + 74740 \, x - 135952\right )} \cos \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \] Input:
integrate(exp(1/2*x)*x^2*cos(x)^3,x, algorithm="fricas")
Output:
12/6331625*(125*(1369*x^2 - 296*x - 264)*cos(x)^2 + 273800*x^2 - 497280*x - 93056)*e^(1/2*x)*sin(x) + 2/6331625*(125*(1369*x^2 + 5180*x - 856)*cos(x )^3 + 24*(34225*x^2 + 74740*x - 135952)*cos(x))*e^(1/2*x)
Time = 0.76 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.08 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {96 x^{2} e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac {48 x^{2} e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{185} + \frac {156 x^{2} e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac {58 x^{2} e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac {32256 x e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac {19392 x e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{34225} - \frac {34656 x e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac {26392 x e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac {1116672 e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac {6525696 e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{6331625} - \frac {1512672 e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac {6739696 e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{6331625} \] Input:
integrate(exp(1/2*x)*x**2*cos(x)**3,x)
Output:
96*x**2*exp(x/2)*sin(x)**3/185 + 48*x**2*exp(x/2)*sin(x)**2*cos(x)/185 + 1 56*x**2*exp(x/2)*sin(x)*cos(x)**2/185 + 58*x**2*exp(x/2)*cos(x)**3/185 - 3 2256*x*exp(x/2)*sin(x)**3/34225 + 19392*x*exp(x/2)*sin(x)**2*cos(x)/34225 - 34656*x*exp(x/2)*sin(x)*cos(x)**2/34225 + 26392*x*exp(x/2)*cos(x)**3/342 25 - 1116672*exp(x/2)*sin(x)**3/6331625 - 6525696*exp(x/2)*sin(x)**2*cos(x )/6331625 - 1512672*exp(x/2)*sin(x)*cos(x)**2/6331625 - 6739696*exp(x/2)*c os(x)**3/6331625
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.41 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{50653} \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac {3}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) \] Input:
integrate(exp(1/2*x)*x^2*cos(x)^3,x, algorithm="maxima")
Output:
1/101306*(1369*x^2 + 5180*x - 856)*cos(3*x)*e^(1/2*x) + 3/250*(25*x^2 + 60 *x - 88)*cos(x)*e^(1/2*x) + 3/50653*(1369*x^2 - 296*x - 264)*e^(1/2*x)*sin (3*x) + 3/125*(25*x^2 - 40*x - 8)*e^(1/2*x)*sin(x)
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {1}{101306} \, {\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \] Input:
integrate(exp(1/2*x)*x^2*cos(x)^3,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) + 3/250*((25*x^2 + 60*x - 88)*cos(x) + 2*(25*x^2 - 40* x - 8)*sin(x))*e^(1/2*x)
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.44 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=-\frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )+13372392\,\cos \left (x\right )+2431344\,\sin \left (x\right )-647500\,x\,\cos \left (3\,x\right )-3798975\,x^2\,\cos \left (x\right )+222000\,x\,\sin \left (3\,x\right )-7597950\,x^2\,\sin \left (x\right )-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )-9117540\,x\,\cos \left (x\right )+12156720\,x\,\sin \left (x\right )\right )}{12663250} \] Input:
int(x^2*exp(x/2)*cos(x)^3,x)
Output:
-(exp(x/2)*(107000*cos(3*x) + 198000*sin(3*x) + 13372392*cos(x) + 2431344* sin(x) - 647500*x*cos(3*x) - 3798975*x^2*cos(x) + 222000*x*sin(3*x) - 7597 950*x^2*sin(x) - 171125*x^2*cos(3*x) - 1026750*x^2*sin(3*x) - 9117540*x*co s(x) + 12156720*x*sin(x)))/12663250
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.48 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {2 e^{\frac {x}{2}} \left (-171125 \cos \left (x \right ) \sin \left (x \right )^{2} x^{2}-647500 \cos \left (x \right ) \sin \left (x \right )^{2} x +107000 \cos \left (x \right ) \sin \left (x \right )^{2}+992525 \cos \left (x \right ) x^{2}+2441260 \cos \left (x \right ) x -3369848 \cos \left (x \right )-1026750 \sin \left (x \right )^{3} x^{2}+222000 \sin \left (x \right )^{3} x +198000 \sin \left (x \right )^{3}+2669550 \sin \left (x \right ) x^{2}-3205680 \sin \left (x \right ) x -756336 \sin \left (x \right )\right )}{6331625} \] Input:
int(exp(1/2*x)*x^2*cos(x)^3,x)
Output:
(2*e**(x/2)*( - 171125*cos(x)*sin(x)**2*x**2 - 647500*cos(x)*sin(x)**2*x + 107000*cos(x)*sin(x)**2 + 992525*cos(x)*x**2 + 2441260*cos(x)*x - 3369848 *cos(x) - 1026750*sin(x)**3*x**2 + 222000*sin(x)**3*x + 198000*sin(x)**3 + 2669550*sin(x)*x**2 - 3205680*sin(x)*x - 756336*sin(x)))/6331625