3.567 \(\int e^{x/2} x^2 \cos ^3(x) \, dx\)

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

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Rubi [A]  time = 0.48, antiderivative size = 253, normalized size of antiderivative = 1.35, number of steps used = 31, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {4435, 4433, 4466, 14, 4432, 4469, 4465, 4470} \[ \frac {96}{185} e^{x/2} x^2 \sin (x)+\frac {2}{37} e^{x/2} x^2 \cos ^3(x)+\frac {48}{185} e^{x/2} x^2 \cos (x)+\frac {12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)-\frac {1218672 e^{x/2} \sin (x)}{6331625}-\frac {32556 e^{x/2} x \sin (x)}{34225}-\frac {816 e^{x/2} \sin (3 x)}{50653}-\frac {12 e^{x/2} x \sin (3 x)}{1369}+\frac {16 e^{x/2} \cos ^3(x)}{50653}-\frac {8 e^{x/2} x \cos ^3(x)}{1369}-\frac {6687696 e^{x/2} \cos (x)}{6331625}+\frac {24792 e^{x/2} x \cos (x)}{34225}-\frac {432 e^{x/2} \cos (3 x)}{50653}+\frac {72 e^{x/2} x \cos (3 x)}{1369}+\frac {96 e^{x/2} \sin (x) \cos ^2(x)}{50653}-\frac {48 e^{x/2} x \sin (x) \cos ^2(x)}{1369} \]

Antiderivative was successfully verified.

[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 4432

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 4433

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 4435

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 4465

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 4466

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 4469

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 4470

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*} \int e^{x/2} x^2 \cos ^3(x) \, 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 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*}

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Mathematica [A]  time = 0.16, size = 72, normalized size = 0.39 \[ \frac {e^{x/2} \left (303918 \left (25 x^2-40 x-8\right ) \sin (x)+750 \left (1369 x^2-296 x-264\right ) \sin (3 x)+151959 \left (25 x^2+60 x-88\right ) \cos (x)+125 \left (1369 x^2+5180 x-856\right ) \cos (3 x)\right )}{12663250} \]

Antiderivative was successfully verified.

[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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x/2} x^2 \cos ^3(x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.33, size = 72, normalized size = 0.39 \[ \frac {12}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \relax (x)^{2} + 273800 \, x^{2} - 497280 \, x - 93056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \relax (x) + \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \relax (x)^{3} + 24 \, {\left (34225 \, x^{2} + 74740 \, x - 135952\right )} \cos \relax (x)\right )} e^{\left (\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.60, size = 73, normalized size = 0.39 \[ \frac {1}{101306} \, {\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \relax (x) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \relax (x)\right )} e^{\left (\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.11, size = 106, normalized size = 0.57

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/202612-3/101306*I)*(888*I*x+1369*x^2-280-96*I-148*x)*exp((1/2+3*I)*x)+(3/500-3/250*I)*(40*I*x+25*x^2-24-32*
I-20*x)*exp((1/2+I)*x)+(3/500+3/250*I)*(-40*I*x+25*x^2-24+32*I-20*x)*exp((1/2-I)*x)+(1/202612+3/101306*I)*(-88
8*I*x+1369*x^2-280+96*I-148*x)*exp((1/2-3*I)*x)

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maxima [A]  time = 0.48, size = 77, normalized size = 0.41 \[ \frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \relax (x) e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{50653} \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac {3}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.31, size = 83, normalized size = 0.44 \[ -\frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )[phone]\,\cos \relax (x)+2431344\,\sin \relax (x)-647500\,x\,\cos \left (3\,x\right )-3798975\,x^2\,\cos \relax (x)+222000\,x\,\sin \left (3\,x\right )-7597950\,x^2\,\sin \relax (x)-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )-9117540\,x\,\cos \relax (x)[phone]\,x\,\sin \relax (x)\right )}{12663250} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 11.77, size = 202, normalized size = 1.08 \[ \frac {96 x^{2} e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{185} + \frac {48 x^{2} e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{185} + \frac {156 x^{2} e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{185} + \frac {58 x^{2} e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{185} - \frac {32256 x e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{34225} + \frac {19392 x e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{34225} - \frac {34656 x e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{34225} + \frac {26392 x e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{34225} - \frac {1116672 e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{6331625} - \frac {6525696 e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{6331625} - \frac {1512672 e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{6331625} - \frac {6739696 e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{6331625} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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