\(\int e^{x/2} x^2 \cos ^3(x) \, dx\) [567]

Optimal result

Integrand size = 15, antiderivative size = 187 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=-\frac {132}{125} e^{x/2} \cos (x)+\frac {18}{25} e^{x/2} x \cos (x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {428 e^{x/2} \cos (3 x)}{50653}+\frac {70 e^{x/2} x \cos (3 x)}{1369}-\frac {24}{125} e^{x/2} \sin (x)-\frac {24}{25} e^{x/2} x \sin (x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-\frac {792 e^{x/2} \sin (3 x)}{50653}-\frac {24 e^{x/2} x \sin (3 x)}{1369} \]

[Out]

-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)-792/50653*exp(1/2*x)*sin(3*x)-24/1369*exp(
1/2*x)*x*sin(3*x)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.35, number of steps used = 31, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {4520, 4518, 4554, 14, 4517, 4557, 4553, 4558} \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)-\frac {1218672 e^{x/2} \sin (x)}{6331625}-\frac {32556 e^{x/2} x \sin (x)}{34225}-\frac {816 e^{x/2} \sin (3 x)}{50653}-\frac {12 e^{x/2} x \sin (3 x)}{1369}+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}-\frac {6687696 e^{x/2} \cos (x)}{6331625}+\frac {24792 e^{x/2} x \cos (x)}{34225}-\frac {432 e^{x/2} \cos (3 x)}{50653}+\frac {72 e^{x/2} x \cos (3 x)}{1369}+\frac {96 e^{x/2} \sin (x) \cos ^2(x)}{50653}-\frac {48 e^{x/2} x \sin (x) \cos ^2(x)}{1369} \]

[In]

Int[E^(x/2)*x^2*Cos[x]^3,x]

[Out]

(-6687696*E^(x/2)*Cos[x])/6331625 + (24792*E^(x/2)*x*Cos[x])/34225 + (48*E^(x/2)*x^2*Cos[x])/185 + (16*E^(x/2)
*Cos[x]^3)/50653 - (8*E^(x/2)*x*Cos[x]^3)/1369 + (2*E^(x/2)*x^2*Cos[x]^3)/37 - (432*E^(x/2)*Cos[3*x])/50653 +
(72*E^(x/2)*x*Cos[3*x])/1369 - (1218672*E^(x/2)*Sin[x])/6331625 - (32556*E^(x/2)*x*Sin[x])/34225 + (96*E^(x/2)
*x^2*Sin[x])/185 + (96*E^(x/2)*Cos[x]^2*Sin[x])/50653 - (48*E^(x/2)*x*Cos[x]^2*Sin[x])/1369 + (12*E^(x/2)*x^2*
Cos[x]^2*Sin[x])/37 - (816*E^(x/2)*Sin[3*x])/50653 - (12*E^(x/2)*x*Sin[3*x])/1369

Rule 14

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]]

Rule 4517

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(S
in[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] - Simp[e*F^(c*(a + b*x))*(Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4518

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(C
os[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4520

Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x
))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + (Dist[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2), Int
[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x] + Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1)/(e
^2*m^2 + b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
m, 1]

Rule 4553

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.)*Sin[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Sin[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rule 4554

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]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rule 4557

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[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]

Rule 4558

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]

Rubi steps \begin{align*} \text {integral}& = \frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-2 \int x \left (\frac {48}{185} e^{x/2} \cos (x)+\frac {2}{37} e^{x/2} \cos ^3(x)+\frac {96}{185} e^{x/2} \sin (x)+\frac {12}{37} e^{x/2} \cos ^2(x) \sin (x)\right ) \, dx \\ & = \frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-2 \int \left (\frac {48}{185} e^{x/2} x \cos (x)+\frac {2}{37} e^{x/2} x \cos ^3(x)+\frac {96}{185} e^{x/2} x \sin (x)+\frac {12}{37} e^{x/2} x \cos ^2(x) \sin (x)\right ) \, dx \\ & = \frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(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 {4}{37} \int e^{x/2} x \cos ^3(x) \, dx-\frac {96}{185} \int e^{x/2} x \cos (x) \, dx-\frac {24}{37} \int e^{x/2} x \cos ^2(x) \sin (x) \, dx-\frac {192}{185} \int e^{x/2} x \sin (x) \, dx \\ & = \frac {20352 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {30336 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+\frac {4}{37} \int \left (\frac {48}{185} e^{x/2} \cos (x)+\frac {2}{37} e^{x/2} \cos ^3(x)+\frac {96}{185} e^{x/2} \sin (x)+\frac {12}{37} e^{x/2} \cos ^2(x) \sin (x)\right ) \, dx+\frac {96}{185} \int \left (\frac {2}{5} e^{x/2} \cos (x)+\frac {4}{5} e^{x/2} \sin (x)\right ) \, dx-\frac {24}{37} \int \left (\frac {1}{4} e^{x/2} x \sin (x)+\frac {1}{4} e^{x/2} x \sin (3 x)\right ) \, dx+\frac {192}{185} \int \left (-\frac {4}{5} e^{x/2} \cos (x)+\frac {2}{5} e^{x/2} \sin (x)\right ) \, dx \\ & = \frac {20352 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {30336 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+\frac {8 \int e^{x/2} \cos ^3(x) \, dx}{1369}+\frac {192 \int e^{x/2} \cos (x) \, dx}{6845}+\frac {48 \int e^{x/2} \cos ^2(x) \sin (x) \, dx}{1369}+\frac {384 \int e^{x/2} \sin (x) \, dx}{6845}-\frac {6}{37} \int e^{x/2} x \sin (x) \, dx-\frac {6}{37} \int e^{x/2} x \sin (3 x) \, dx+\frac {192}{925} \int e^{x/2} \cos (x) \, dx+2 \left (\frac {384}{925} \int e^{x/2} \sin (x) \, dx\right )-\frac {768}{925} \int e^{x/2} \cos (x) \, dx \\ & = -\frac {48384 e^{x/2} \cos (x)}{171125}+\frac {24792 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {72 e^{x/2} x \cos (3 x)}{1369}-\frac {77568 e^{x/2} \sin (x)}{171125}-\frac {32556 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac {1536 e^{x/2} \cos (x)}{4625}+\frac {768 e^{x/2} \sin (x)}{4625}\right )-\frac {12 e^{x/2} x \sin (3 x)}{1369}+\frac {192 \int e^{x/2} \cos (x) \, dx}{50653}+\frac {48 \int \left (\frac {1}{4} e^{x/2} \sin (x)+\frac {1}{4} e^{x/2} \sin (3 x)\right ) \, dx}{1369}+\frac {6}{37} \int \left (-\frac {4}{5} e^{x/2} \cos (x)+\frac {2}{5} e^{x/2} \sin (x)\right ) \, dx+\frac {6}{37} \int \left (-\frac {12}{37} e^{x/2} \cos (3 x)+\frac {2}{37} e^{x/2} \sin (3 x)\right ) \, dx \\ & = -\frac {1780608 e^{x/2} \cos (x)}{6331625}+\frac {24792 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {72 e^{x/2} x \cos (3 x)}{1369}-\frac {2850816 e^{x/2} \sin (x)}{6331625}-\frac {32556 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac {1536 e^{x/2} \cos (x)}{4625}+\frac {768 e^{x/2} \sin (x)}{4625}\right )-\frac {12 e^{x/2} x \sin (3 x)}{1369}+\frac {12 \int e^{x/2} \sin (x) \, dx}{1369}+2 \frac {12 \int e^{x/2} \sin (3 x) \, dx}{1369}-\frac {72 \int e^{x/2} \cos (3 x) \, dx}{1369}+\frac {12}{185} \int e^{x/2} \sin (x) \, dx-\frac {24}{185} \int e^{x/2} \cos (x) \, dx \\ & = -\frac {2482128 e^{x/2} \cos (x)}{6331625}+\frac {24792 e^{x/2} x \cos (x)}{34225}+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)-\frac {144 e^{x/2} \cos (3 x)}{50653}+\frac {72 e^{x/2} x \cos (3 x)}{1369}-\frac {3321456 e^{x/2} \sin (x)}{6331625}-\frac {32556 e^{x/2} x \sin (x)}{34225}+\frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac {48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac {12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac {1536 e^{x/2} \cos (x)}{4625}+\frac {768 e^{x/2} \sin (x)}{4625}\right )-\frac {864 e^{x/2} \sin (3 x)}{50653}-\frac {12 e^{x/2} x \sin (3 x)}{1369}+2 \left (-\frac {144 e^{x/2} \cos (3 x)}{50653}+\frac {24 e^{x/2} \sin (3 x)}{50653}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {e^{x/2} \left (151959 \left (-88+60 x+25 x^2\right ) \cos (x)+125 \left (-856+5180 x+1369 x^2\right ) \cos (3 x)+303918 \left (-8-40 x+25 x^2\right ) \sin (x)+750 \left (-264-296 x+1369 x^2\right ) \sin (3 x)\right )}{12663250} \]

[In]

Integrate[E^(x/2)*x^2*Cos[x]^3,x]

[Out]

(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 + 1369*x^2)*Sin[3*x]))/12663250

Maple [A] (verified)

Time = 0.34 (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	\(\left (\frac {1}{202612}-\frac {3 i}{101306}\right ) \left (1369 x^{2}+888 i x -148 x -96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}+3 i\right ) x}+\left (\frac {3}{500}-\frac {3 i}{250}\right ) \left (25 x^{2}+40 i x -20 x -32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}+i\right ) x}+\left (\frac {3}{500}+\frac {3 i}{250}\right ) \left (25 x^{2}-40 i x -20 x +32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}-i\right ) x}+\left (\frac {1}{202612}+\frac {3 i}{101306}\right ) \left (1369 x^{2}-888 i x -148 x +96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}-3 i\right ) x}\)	\(106\)

[In]

int(exp(1/2*x)*x^2*cos(x)^3,x,method=_RETURNVERBOSE)

[Out]

3/10*(5/111*(x^2+140/37*x-856/1369)*cos(3*x)+10/37*(x^2-8/37*x-264/1369)*sin(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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {12}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} + 273800 \, x^{2} - 497280 \, x - 93056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) + \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (x\right )^{3} + 24 \, {\left (34225 \, x^{2} + 74740 \, x - 135952\right )} \cos \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \]

[In]

integrate(exp(1/2*x)*x^2*cos(x)^3,x, algorithm="fricas")

[Out]

12/6331625*(125*(1369*x^2 - 296*x - 264)*cos(x)^2 + 273800*x^2 - 497280*x - 93056)*e^(1/2*x)*sin(x) + 2/633162
5*(125*(1369*x^2 + 5180*x - 856)*cos(x)^3 + 24*(34225*x^2 + 74740*x - 135952)*cos(x))*e^(1/2*x)

Sympy [A] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.08 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {96 x^{2} e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac {48 x^{2} e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{185} + \frac {156 x^{2} e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac {58 x^{2} e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac {32256 x e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac {19392 x e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{34225} - \frac {34656 x e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac {26392 x e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac {1116672 e^{\frac {x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac {6525696 e^{\frac {x}{2}} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{6331625} - \frac {1512672 e^{\frac {x}{2}} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac {6739696 e^{\frac {x}{2}} \cos ^{3}{\left (x \right )}}{6331625} \]

[In]

integrate(exp(1/2*x)*x**2*cos(x)**3,x)

[Out]

96*x**2*exp(x/2)*sin(x)**3/185 + 48*x**2*exp(x/2)*sin(x)**2*cos(x)/185 + 156*x**2*exp(x/2)*sin(x)*cos(x)**2/18
5 + 58*x**2*exp(x/2)*cos(x)**3/185 - 32256*x*exp(x/2)*sin(x)**3/34225 + 19392*x*exp(x/2)*sin(x)**2*cos(x)/3422
5 - 34656*x*exp(x/2)*sin(x)*cos(x)**2/34225 + 26392*x*exp(x/2)*cos(x)**3/34225 - 1116672*exp(x/2)*sin(x)**3/63
31625 - 6525696*exp(x/2)*sin(x)**2*cos(x)/6331625 - 1512672*exp(x/2)*sin(x)*cos(x)**2/6331625 - 6739696*exp(x/
2)*cos(x)**3/6331625

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.41 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{50653} \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac {3}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (x\right ) \]

[In]

integrate(exp(1/2*x)*x^2*cos(x)^3,x, algorithm="maxima")

[Out]

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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.39 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=\frac {1}{101306} \, {\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \left (x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} \]

[In]

integrate(exp(1/2*x)*x^2*cos(x)^3,x, algorithm="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.44 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx=-\frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )[phone]\,\cos \left (x\right )+2431344\,\sin \left (x\right )-647500\,x\,\cos \left (3\,x\right )-3798975\,x^2\,\cos \left (x\right )+222000\,x\,\sin \left (3\,x\right )-7597950\,x^2\,\sin \left (x\right )-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )-9117540\,x\,\cos \left (x\right )[phone]\,x\,\sin \left (x\right )\right )}{12663250} \]

[In]

int(x^2*exp(x/2)*cos(x)^3,x)

[Out]

-(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) - 7597950*x^2*sin(x) - 171125*x^2*cos(3*x) - 1026750*x^2*sin(3*x) - 9117540*x*
cos(x) + 12156720*x*sin(x)))/12663250