Integrand size = 8, antiderivative size = 83 \[ \int x^2 \cos ^5(x) \, dx=\frac {16}{15} x \cos (x)+\frac {8}{45} x \cos ^3(x)+\frac {2}{25} x \cos ^5(x)-\frac {298 \sin (x)}{225}+\frac {8}{15} x^2 \sin (x)+\frac {4}{15} x^2 \cos ^2(x) \sin (x)+\frac {1}{5} x^2 \cos ^4(x) \sin (x)+\frac {76 \sin ^3(x)}{675}-\frac {2 \sin ^5(x)}{125} \]
16/15*x*cos(x)+8/45*x*cos(x)^3+2/25*x*cos(x)^5-298/225*sin(x)+8/15*x^2*sin (x)+4/15*x^2*cos(x)^2*sin(x)+1/5*x^2*cos(x)^4*sin(x)+76/675*sin(x)^3-2/125 *sin(x)^5
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int x^2 \cos ^5(x) \, dx=\frac {5}{4} x \cos (x)+\frac {5}{72} x \cos (3 x)+\frac {1}{200} x \cos (5 x)+\frac {5}{8} \left (-2+x^2\right ) \sin (x)+\frac {5}{432} \left (-2+9 x^2\right ) \sin (3 x)+\frac {\left (-2+25 x^2\right ) \sin (5 x)}{2000} \]
(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])/432 + ((-2 + 25*x^2)*Sin[5*x])/2000
Time = 0.62 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.875, Rules used = {3042, 3792, 3042, 3113, 2009, 3792, 3042, 3113, 2009, 3777, 25, 3042, 3777, 3042, 3117}
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.
| \(\displaystyle \int x^2 \cos ^5(x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^2 \sin \left (x+\frac {\pi }{2}\right )^5dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {4}{5} \int x^2 \cos ^3(x)dx-\frac {2}{25} \int \cos ^5(x)dx+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{5} \int x^2 \sin \left (x+\frac {\pi }{2}\right )^3dx-\frac {2}{25} \int \sin \left (x+\frac {\pi }{2}\right )^5dx+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {4}{5} \int x^2 \sin \left (x+\frac {\pi }{2}\right )^3dx+\frac {2}{25} \int \left (\sin ^4(x)-2 \sin ^2(x)+1\right )d(-\sin (x))+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{5} \int x^2 \sin \left (x+\frac {\pi }{2}\right )^3dx+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int x^2 \cos (x)dx-\frac {2}{9} \int \cos ^3(x)dx+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int x^2 \sin \left (x+\frac {\pi }{2}\right )dx-\frac {2}{9} \int \sin \left (x+\frac {\pi }{2}\right )^3dx+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int x^2 \sin \left (x+\frac {\pi }{2}\right )dx+\frac {2}{9} \int \left (1-\sin ^2(x)\right )d(-\sin (x))+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int x^2 \sin \left (x+\frac {\pi }{2}\right )dx+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (2 \int -x \sin (x)dx+x^2 \sin (x)\right )+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x^2 \sin (x)-2 \int x \sin (x)dx\right )+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x^2 \sin (x)-2 \int x \sin (x)dx\right )+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x^2 \sin (x)-2 (\int \cos (x)dx-x \cos (x))\right )+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x^2 \sin (x)-2 \left (\int \sin \left (x+\frac {\pi }{2}\right )dx-x \cos (x)\right )\right )+\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{9} \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{9} x \cos ^3(x)\right )+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {4}{5} \left (\frac {1}{3} x^2 \sin (x) \cos ^2(x)+\frac {2}{3} \left (x^2 \sin (x)-2 (\sin (x)-x \cos (x))\right )+\frac {2}{9} \left (\frac {\sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{9} x \cos ^3(x)\right )+\frac {2}{25} \left (-\frac {1}{5} \sin ^5(x)+\frac {2 \sin ^3(x)}{3}-\sin (x)\right )+\frac {2}{25} x \cos ^5(x)\) |
(2*x*Cos[x]^5)/25 + (x^2*Cos[x]^4*Sin[x])/5 + (2*(-Sin[x] + (2*Sin[x]^3)/3 - Sin[x]^5/5))/25 + (4*((2*x*Cos[x]^3)/9 + (x^2*Cos[x]^2*Sin[x])/3 + (2*( -Sin[x] + Sin[x]^3/3))/9 + (2*(x^2*Sin[x] - 2*(-(x*Cos[x]) + Sin[x])))/3)) /5
3.5.83.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Time = 0.53 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\frac {5 x \cos \left (x \right )}{4}+\frac {5 \left (x^{2}-2\right ) \sin \left (x \right )}{8}+\frac {x \cos \left (5 x \right )}{200}+\frac {\left (25 x^{2}-2\right ) \sin \left (5 x \right )}{2000}+\frac {5 x \cos \left (3 x \right )}{72}+\frac {5 \left (9 x^{2}-2\right ) \sin \left (3 x \right )}{432}\) | \(56\) |
| parallelrisch | \(\frac {\left (5625 x^{2}-1250\right ) \sin \left (3 x \right )}{54000}+\frac {\left (675 x^{2}-54\right ) \sin \left (5 x \right )}{54000}+\frac {5 x^{2} \sin \left (x \right )}{8}+\frac {5 x \cos \left (x \right )}{4}+\frac {5 x \cos \left (3 x \right )}{72}+\frac {x \cos \left (5 x \right )}{200}-\frac {5 \sin \left (x \right )}{4}\) | \(58\) |
| default | \(\frac {x^{2} \left (\frac {8}{3}+\cos ^{4}\left (x \right )+\frac {4 \left (\cos ^{2}\left (x \right )\right )}{3}\right ) \sin \left (x \right )}{5}+\frac {2 \left (\cos ^{5}\left (x \right )\right ) x}{25}-\frac {2 \left (\frac {8}{3}+\cos ^{4}\left (x \right )+\frac {4 \left (\cos ^{2}\left (x \right )\right )}{3}\right ) \sin \left (x \right )}{125}+\frac {8 x \left (\cos ^{3}\left (x \right )\right )}{45}-\frac {8 \left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{135}-\frac {16 \sin \left (x \right )}{15}+\frac {16 x \cos \left (x \right )}{15}\) | \(70\) |
5/4*x*cos(x)+5/8*(x^2-2)*sin(x)+1/200*x*cos(5*x)+1/2000*(25*x^2-2)*sin(5*x )+5/72*x*cos(3*x)+5/432*(9*x^2-2)*sin(3*x)
Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.69 \[ \int x^2 \cos ^5(x) \, dx=\frac {2}{25} \, x \cos \left (x\right )^{5} + \frac {8}{45} \, x \cos \left (x\right )^{3} + \frac {16}{15} \, x \cos \left (x\right ) + \frac {1}{3375} \, {\left (27 \, {\left (25 \, x^{2} - 2\right )} \cos \left (x\right )^{4} + 4 \, {\left (225 \, x^{2} - 68\right )} \cos \left (x\right )^{2} + 1800 \, x^{2} - 4144\right )} \sin \left (x\right ) \]
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)
Time = 0.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35 \[ \int x^2 \cos ^5(x) \, dx=\frac {8 x^{2} \sin ^{5}{\left (x \right )}}{15} + \frac {4 x^{2} \sin ^{3}{\left (x \right )} \cos ^{2}{\left (x \right )}}{3} + x^{2} \sin {\left (x \right )} \cos ^{4}{\left (x \right )} + \frac {16 x \sin ^{4}{\left (x \right )} \cos {\left (x \right )}}{15} + \frac {104 x \sin ^{2}{\left (x \right )} \cos ^{3}{\left (x \right )}}{45} + \frac {298 x \cos ^{5}{\left (x \right )}}{225} - \frac {4144 \sin ^{5}{\left (x \right )}}{3375} - \frac {1712 \sin ^{3}{\left (x \right )} \cos ^{2}{\left (x \right )}}{675} - \frac {298 \sin {\left (x \right )} \cos ^{4}{\left (x \right )}}{225} \]
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)*c os(x)**4/225
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.66 \[ \int x^2 \cos ^5(x) \, dx=\frac {1}{200} \, x \cos \left (5 \, x\right ) + \frac {5}{72} \, x \cos \left (3 \, x\right ) + \frac {5}{4} \, x \cos \left (x\right ) + \frac {1}{2000} \, {\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac {5}{432} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {5}{8} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \]
1/200*x*cos(5*x) + 5/72*x*cos(3*x) + 5/4*x*cos(x) + 1/2000*(25*x^2 - 2)*si n(5*x) + 5/432*(9*x^2 - 2)*sin(3*x) + 5/8*(x^2 - 2)*sin(x)
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.66 \[ \int x^2 \cos ^5(x) \, dx=\frac {1}{200} \, x \cos \left (5 \, x\right ) + \frac {5}{72} \, x \cos \left (3 \, x\right ) + \frac {5}{4} \, x \cos \left (x\right ) + \frac {1}{2000} \, {\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac {5}{432} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {5}{8} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \]
1/200*x*cos(5*x) + 5/72*x*cos(3*x) + 5/4*x*cos(x) + 1/2000*(25*x^2 - 2)*si n(5*x) + 5/432*(9*x^2 - 2)*sin(3*x) + 5/8*(x^2 - 2)*sin(x)
Time = 0.42 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int x^2 \cos ^5(x) \, dx=\frac {8\,x\,{\cos \left (x\right )}^3}{45}-\frac {4144\,\sin \left (x\right )}{3375}+\frac {2\,x\,{\cos \left (x\right )}^5}{25}+\frac {8\,x^2\,\sin \left (x\right )}{15}-\frac {272\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{3375}-\frac {2\,{\cos \left (x\right )}^4\,\sin \left (x\right )}{125}+\frac {16\,x\,\cos \left (x\right )}{15}+\frac {4\,x^2\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{15}+\frac {x^2\,{\cos \left (x\right )}^4\,\sin \left (x\right )}{5} \]