3.483 \(\int x^2 \cos ^5(x) \, dx\)

Optimal. Leaf size=83 \[ \frac{8}{15} x^2 \sin (x)+\frac{1}{5} x^2 \sin (x) \cos ^4(x)+\frac{4}{15} x^2 \sin (x) \cos ^2(x)-\frac{2 \sin ^5(x)}{125}+\frac{76 \sin ^3(x)}{675}-\frac{298 \sin (x)}{225}+\frac{2}{25} x \cos ^5(x)+\frac{8}{45} x \cos ^3(x)+\frac{16}{15} x \cos (x) \]

[Out]

(16*x*Cos[x])/15 + (8*x*Cos[x]^3)/45 + (2*x*Cos[x]^5)/25 - (298*Sin[x])/225 + (8*x^2*Sin[x])/15 + (4*x^2*Cos[x
]^2*Sin[x])/15 + (x^2*Cos[x]^4*Sin[x])/5 + (76*Sin[x]^3)/675 - (2*Sin[x]^5)/125

________________________________________________________________________________________

Rubi [A]  time = 0.0937162, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3311, 3296, 2637, 2633} \[ \frac{8}{15} x^2 \sin (x)+\frac{1}{5} x^2 \sin (x) \cos ^4(x)+\frac{4}{15} x^2 \sin (x) \cos ^2(x)-\frac{2 \sin ^5(x)}{125}+\frac{76 \sin ^3(x)}{675}-\frac{298 \sin (x)}{225}+\frac{2}{25} x \cos ^5(x)+\frac{8}{45} x \cos ^3(x)+\frac{16}{15} x \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[x]^5,x]

[Out]

(16*x*Cos[x])/15 + (8*x*Cos[x]^3)/45 + (2*x*Cos[x]^5)/25 - (298*Sin[x])/225 + (8*x^2*Sin[x])/15 + (4*x^2*Cos[x
]^2*Sin[x])/15 + (x^2*Cos[x]^4*Sin[x])/5 + (76*Sin[x]^3)/675 - (2*Sin[x]^5)/125

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int x^2 \cos ^5(x) \, dx &=\frac{2}{25} x \cos ^5(x)+\frac{1}{5} x^2 \cos ^4(x) \sin (x)-\frac{2}{25} \int \cos ^5(x) \, dx+\frac{4}{5} \int x^2 \cos ^3(x) \, dx\\ &=\frac{8}{45} x \cos ^3(x)+\frac{2}{25} x \cos ^5(x)+\frac{4}{15} x^2 \cos ^2(x) \sin (x)+\frac{1}{5} x^2 \cos ^4(x) \sin (x)+\frac{2}{25} \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )-\frac{8}{45} \int \cos ^3(x) \, dx+\frac{8}{15} \int x^2 \cos (x) \, dx\\ &=\frac{8}{45} x \cos ^3(x)+\frac{2}{25} x \cos ^5(x)-\frac{2 \sin (x)}{25}+\frac{8}{15} x^2 \sin (x)+\frac{4}{15} x^2 \cos ^2(x) \sin (x)+\frac{1}{5} x^2 \cos ^4(x) \sin (x)+\frac{4 \sin ^3(x)}{75}-\frac{2 \sin ^5(x)}{125}+\frac{8}{45} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )-\frac{16}{15} \int x \sin (x) \, dx\\ &=\frac{16}{15} x \cos (x)+\frac{8}{45} x \cos ^3(x)+\frac{2}{25} x \cos ^5(x)-\frac{58 \sin (x)}{225}+\frac{8}{15} x^2 \sin (x)+\frac{4}{15} x^2 \cos ^2(x) \sin (x)+\frac{1}{5} x^2 \cos ^4(x) \sin (x)+\frac{76 \sin ^3(x)}{675}-\frac{2 \sin ^5(x)}{125}-\frac{16}{15} \int \cos (x) \, dx\\ &=\frac{16}{15} x \cos (x)+\frac{8}{45} x \cos ^3(x)+\frac{2}{25} x \cos ^5(x)-\frac{298 \sin (x)}{225}+\frac{8}{15} x^2 \sin (x)+\frac{4}{15} x^2 \cos ^2(x) \sin (x)+\frac{1}{5} x^2 \cos ^4(x) \sin (x)+\frac{76 \sin ^3(x)}{675}-\frac{2 \sin ^5(x)}{125}\\ \end{align*}

Mathematica [A]  time = 0.0539509, size = 67, normalized size = 0.81 \[ \frac{5}{8} \left (x^2-2\right ) \sin (x)+\frac{5}{432} \left (9 x^2-2\right ) \sin (3 x)+\frac{\left (25 x^2-2\right ) \sin (5 x)}{2000}+\frac{5}{4} x \cos (x)+\frac{5}{72} x \cos (3 x)+\frac{1}{200} x \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[x]^5,x]

[Out]

(5*x*Cos[x])/4 + (5*x*Cos[3*x])/72 + (x*Cos[5*x])/200 + (5*(-2 + x^2)*Sin[x])/8 + (5*(-2 + 9*x^2)*Sin[3*x])/43
2 + ((-2 + 25*x^2)*Sin[5*x])/2000

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 70, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}\sin \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) }-{\frac{16\,\sin \left ( x \right ) }{15}}+{\frac{16\,x\cos \left ( x \right ) }{15}}+{\frac{2\,x \left ( \cos \left ( x \right ) \right ) ^{5}}{25}}-{\frac{2\,\sin \left ( x \right ) }{125} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) }+{\frac{8\,x \left ( \cos \left ( x \right ) \right ) ^{3}}{45}}-{\frac{ \left ( 16+8\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{135}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(x)^5,x)

[Out]

1/5*x^2*(8/3+cos(x)^4+4/3*cos(x)^2)*sin(x)-16/15*sin(x)+16/15*x*cos(x)+2/25*x*cos(x)^5-2/125*(8/3+cos(x)^4+4/3
*cos(x)^2)*sin(x)+8/45*x*cos(x)^3-8/135*(2+cos(x)^2)*sin(x)

________________________________________________________________________________________

Maxima [A]  time = 0.960549, size = 74, normalized size = 0.89 \begin{align*} \frac{1}{200} \, x \cos \left (5 \, x\right ) + \frac{5}{72} \, x \cos \left (3 \, x\right ) + \frac{5}{4} \, x \cos \left (x\right ) + \frac{1}{2000} \,{\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac{5}{432} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{5}{8} \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(x)^5,x, algorithm="maxima")

[Out]

1/200*x*cos(5*x) + 5/72*x*cos(3*x) + 5/4*x*cos(x) + 1/2000*(25*x^2 - 2)*sin(5*x) + 5/432*(9*x^2 - 2)*sin(3*x)
+ 5/8*(x^2 - 2)*sin(x)

________________________________________________________________________________________

Fricas [A]  time = 2.31121, size = 190, normalized size = 2.29 \begin{align*} \frac{2}{25} \, x \cos \left (x\right )^{5} + \frac{8}{45} \, x \cos \left (x\right )^{3} + \frac{16}{15} \, x \cos \left (x\right ) + \frac{1}{3375} \,{\left (27 \,{\left (25 \, x^{2} - 2\right )} \cos \left (x\right )^{4} + 4 \,{\left (225 \, x^{2} - 68\right )} \cos \left (x\right )^{2} + 1800 \, x^{2} - 4144\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(x)^5,x, algorithm="fricas")

[Out]

2/25*x*cos(x)^5 + 8/45*x*cos(x)^3 + 16/15*x*cos(x) + 1/3375*(27*(25*x^2 - 2)*cos(x)^4 + 4*(225*x^2 - 68)*cos(x
)^2 + 1800*x^2 - 4144)*sin(x)

________________________________________________________________________________________

Sympy [A]  time = 3.46872, size = 112, normalized size = 1.35 \begin{align*} \frac{8 x^{2} \sin ^{5}{\left (x \right )}}{15} + \frac{4 x^{2} \sin ^{3}{\left (x \right )} \cos ^{2}{\left (x \right )}}{3} + x^{2} \sin{\left (x \right )} \cos ^{4}{\left (x \right )} + \frac{16 x \sin ^{4}{\left (x \right )} \cos{\left (x \right )}}{15} + \frac{104 x \sin ^{2}{\left (x \right )} \cos ^{3}{\left (x \right )}}{45} + \frac{298 x \cos ^{5}{\left (x \right )}}{225} - \frac{4144 \sin ^{5}{\left (x \right )}}{3375} - \frac{1712 \sin ^{3}{\left (x \right )} \cos ^{2}{\left (x \right )}}{675} - \frac{298 \sin{\left (x \right )} \cos ^{4}{\left (x \right )}}{225} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cos(x)**5,x)

[Out]

8*x**2*sin(x)**5/15 + 4*x**2*sin(x)**3*cos(x)**2/3 + x**2*sin(x)*cos(x)**4 + 16*x*sin(x)**4*cos(x)/15 + 104*x*
sin(x)**2*cos(x)**3/45 + 298*x*cos(x)**5/225 - 4144*sin(x)**5/3375 - 1712*sin(x)**3*cos(x)**2/675 - 298*sin(x)
*cos(x)**4/225

________________________________________________________________________________________

Giac [A]  time = 1.07878, size = 74, normalized size = 0.89 \begin{align*} \frac{1}{200} \, x \cos \left (5 \, x\right ) + \frac{5}{72} \, x \cos \left (3 \, x\right ) + \frac{5}{4} \, x \cos \left (x\right ) + \frac{1}{2000} \,{\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac{5}{432} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac{5}{8} \,{\left (x^{2} - 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(x)^5,x, algorithm="giac")

[Out]

1/200*x*cos(5*x) + 5/72*x*cos(3*x) + 5/4*x*cos(x) + 1/2000*(25*x^2 - 2)*sin(5*x) + 5/432*(9*x^2 - 2)*sin(3*x)
+ 5/8*(x^2 - 2)*sin(x)