Integrand size = 8, antiderivative size = 105 \[ \int x^2 \sin ^6(x) \, dx=-\frac {245 x}{1152}+\frac {5 x^3}{48}+\frac {245 \cos (x) \sin (x)}{1152}-\frac {5}{16} x^2 \cos (x) \sin (x)+\frac {5}{16} x \sin ^2(x)+\frac {65 \cos (x) \sin ^3(x)}{1728}-\frac {5}{24} x^2 \cos (x) \sin ^3(x)+\frac {5}{48} x \sin ^4(x)+\frac {1}{108} \cos (x) \sin ^5(x)-\frac {1}{6} x^2 \cos (x) \sin ^5(x)+\frac {1}{18} x \sin ^6(x) \]
-245/1152*x+5/48*x^3+245/1152*cos(x)*sin(x)-5/16*x^2*cos(x)*sin(x)+5/16*x* sin(x)^2+65/1728*cos(x)*sin(x)^3-5/24*x^2*cos(x)*sin(x)^3+5/48*x*sin(x)^4+ 1/108*cos(x)*sin(x)^5-1/6*x^2*cos(x)*sin(x)^5+1/18*x*sin(x)^6
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int x^2 \sin ^6(x) \, dx=\frac {1440 x^3-3240 x \cos (2 x)+324 x \cos (4 x)-24 x \cos (6 x)-1620 \left (-1+2 x^2\right ) \sin (2 x)+81 \left (-1+8 x^2\right ) \sin (4 x)-4 \left (-1+18 x^2\right ) \sin (6 x)}{13824} \]
(1440*x^3 - 3240*x*Cos[2*x] + 324*x*Cos[4*x] - 24*x*Cos[6*x] - 1620*(-1 + 2*x^2)*Sin[2*x] + 81*(-1 + 8*x^2)*Sin[4*x] - 4*(-1 + 18*x^2)*Sin[6*x])/138 24
Time = 0.77 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.72, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.500, Rules used = {3042, 3792, 3042, 3115, 3042, 3115, 3042, 3115, 24, 3792, 3042, 3115, 3042, 3115, 24, 3792, 15, 3042, 3115, 24}
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.
| \(\displaystyle \int x^2 \sin ^6(x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^2 \sin (x)^6dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin ^4(x)dx-\frac {1}{18} \int \sin ^6(x)dx-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin (x)^4dx-\frac {1}{18} \int \sin (x)^6dx-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin (x)^4dx+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \int \sin ^4(x)dx\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin (x)^4dx+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \int \sin (x)^4dx\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin (x)^4dx+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \int \sin ^2(x)dx-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin (x)^4dx+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \int \sin (x)^2dx-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin (x)^4dx+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {5}{6} \int x^2 \sin (x)^4dx-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int x^2 \sin ^2(x)dx-\frac {1}{8} \int \sin ^4(x)dx-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int x^2 \sin (x)^2dx-\frac {1}{8} \int \sin (x)^4dx-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int x^2 \sin (x)^2dx+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \int \sin ^2(x)dx\right )-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int x^2 \sin (x)^2dx+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \int \sin (x)^2dx\right )-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int x^2 \sin (x)^2dx+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int x^2 \sin (x)^2dx-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {\int x^2dx}{2}-\frac {1}{2} \int \sin ^2(x)dx-\frac {1}{2} x^2 \sin (x) \cos (x)+\frac {1}{2} x \sin ^2(x)\right )-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {1}{2} \int \sin ^2(x)dx+\frac {x^3}{6}-\frac {1}{2} x^2 \sin (x) \cos (x)+\frac {1}{2} x \sin ^2(x)\right )-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {1}{2} \int \sin (x)^2dx+\frac {x^3}{6}-\frac {1}{2} x^2 \sin (x) \cos (x)+\frac {1}{2} x \sin ^2(x)\right )-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \left (\frac {1}{2} \sin (x) \cos (x)-\frac {\int 1dx}{2}\right )+\frac {x^3}{6}-\frac {1}{2} x^2 \sin (x) \cos (x)+\frac {1}{2} x \sin ^2(x)\right )-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {1}{8} x \sin ^4(x)+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )-\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {1}{6} x^2 \sin ^5(x) \cos (x)+\frac {5}{6} \left (-\frac {1}{4} x^2 \sin ^3(x) \cos (x)+\frac {3}{4} \left (\frac {x^3}{6}-\frac {1}{2} x^2 \sin (x) \cos (x)+\frac {1}{2} x \sin ^2(x)+\frac {1}{2} \left (\frac {1}{2} \sin (x) \cos (x)-\frac {x}{2}\right )\right )+\frac {1}{8} x \sin ^4(x)+\frac {1}{8} \left (\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )+\frac {1}{18} x \sin ^6(x)+\frac {1}{18} \left (\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )\right )\) |
-1/6*(x^2*Cos[x]*Sin[x]^5) + (x*Sin[x]^6)/18 + ((Cos[x]*Sin[x]^5)/6 - (5*( -1/4*(Cos[x]*Sin[x]^3) + (3*(x/2 - (Cos[x]*Sin[x])/2))/4))/6)/18 + (5*(-1/ 4*(x^2*Cos[x]*Sin[x]^3) + (x*Sin[x]^4)/8 + ((Cos[x]*Sin[x]^3)/4 - (3*(x/2 - (Cos[x]*Sin[x])/2))/4)/8 + (3*(x^3/6 - (x^2*Cos[x]*Sin[x])/2 + (x*Sin[x] ^2)/2 + (-1/2*x + (Cos[x]*Sin[x])/2)/2))/4))/6
3.5.85.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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 = 0.84 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {5 x^{3}}{48}-\frac {x \cos \left (6 x \right )}{576}-\frac {\left (18 x^{2}-1\right ) \sin \left (6 x \right )}{3456}+\frac {3 x \cos \left (4 x \right )}{128}+\frac {3 \left (8 x^{2}-1\right ) \sin \left (4 x \right )}{512}-\frac {15 x \cos \left (2 x \right )}{64}-\frac {15 \left (2 x^{2}-1\right ) \sin \left (2 x \right )}{128}\) | \(67\) |
| default | \(x^{2} \left (-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+\frac {x \left (\sin ^{6}\left (x \right )\right )}{18}+\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{108}+\frac {115 x}{1152}+\frac {5 x \left (\sin ^{4}\left (x \right )\right )}{48}+\frac {5 \left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{192}-\frac {5 \left (\cos ^{2}\left (x \right )\right ) x}{16}+\frac {5 \cos \left (x \right ) \sin \left (x \right )}{32}-\frac {5 x^{3}}{24}\) | \(96\) |
5/48*x^3-1/576*x*cos(6*x)-1/3456*(18*x^2-1)*sin(6*x)+3/128*x*cos(4*x)+3/51 2*(8*x^2-1)*sin(4*x)-15/64*x*cos(2*x)-15/128*(2*x^2-1)*sin(2*x)
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.69 \[ \int x^2 \sin ^6(x) \, dx=-\frac {1}{18} \, x \cos \left (x\right )^{6} + \frac {13}{48} \, x \cos \left (x\right )^{4} + \frac {5}{48} \, x^{3} - \frac {11}{16} \, x \cos \left (x\right )^{2} - \frac {1}{3456} \, {\left (32 \, {\left (18 \, x^{2} - 1\right )} \cos \left (x\right )^{5} - 2 \, {\left (936 \, x^{2} - 97\right )} \cos \left (x\right )^{3} + 3 \, {\left (792 \, x^{2} - 299\right )} \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {299}{1152} \, x \]
-1/18*x*cos(x)^6 + 13/48*x*cos(x)^4 + 5/48*x^3 - 11/16*x*cos(x)^2 - 1/3456 *(32*(18*x^2 - 1)*cos(x)^5 - 2*(936*x^2 - 97)*cos(x)^3 + 3*(792*x^2 - 299) *cos(x))*sin(x) + 299/1152*x
Time = 0.55 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.83 \[ \int x^2 \sin ^6(x) \, dx=\frac {5 x^{3} \sin ^{6}{\left (x \right )}}{48} + \frac {5 x^{3} \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {5 x^{3} \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + \frac {5 x^{3} \cos ^{6}{\left (x \right )}}{48} - \frac {11 x^{2} \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{16} - \frac {5 x^{2} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac {5 x^{2} \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{16} + \frac {299 x \sin ^{6}{\left (x \right )}}{1152} + \frac {35 x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{384} - \frac {125 x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{384} - \frac {245 x \cos ^{6}{\left (x \right )}}{1152} + \frac {299 \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{1152} + \frac {25 \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{54} + \frac {245 \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{1152} \]
5*x**3*sin(x)**6/48 + 5*x**3*sin(x)**4*cos(x)**2/16 + 5*x**3*sin(x)**2*cos (x)**4/16 + 5*x**3*cos(x)**6/48 - 11*x**2*sin(x)**5*cos(x)/16 - 5*x**2*sin (x)**3*cos(x)**3/6 - 5*x**2*sin(x)*cos(x)**5/16 + 299*x*sin(x)**6/1152 + 3 5*x*sin(x)**4*cos(x)**2/384 - 125*x*sin(x)**2*cos(x)**4/384 - 245*x*cos(x) **6/1152 + 299*sin(x)**5*cos(x)/1152 + 25*sin(x)**3*cos(x)**3/54 + 245*sin (x)*cos(x)**5/1152
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int x^2 \sin ^6(x) \, dx=\frac {5}{48} \, x^{3} - \frac {1}{576} \, x \cos \left (6 \, x\right ) + \frac {3}{128} \, x \cos \left (4 \, x\right ) - \frac {15}{64} \, x \cos \left (2 \, x\right ) - \frac {1}{3456} \, {\left (18 \, x^{2} - 1\right )} \sin \left (6 \, x\right ) + \frac {3}{512} \, {\left (8 \, x^{2} - 1\right )} \sin \left (4 \, x\right ) - \frac {15}{128} \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]
5/48*x^3 - 1/576*x*cos(6*x) + 3/128*x*cos(4*x) - 15/64*x*cos(2*x) - 1/3456 *(18*x^2 - 1)*sin(6*x) + 3/512*(8*x^2 - 1)*sin(4*x) - 15/128*(2*x^2 - 1)*s in(2*x)
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int x^2 \sin ^6(x) \, dx=\frac {5}{48} \, x^{3} - \frac {1}{576} \, x \cos \left (6 \, x\right ) + \frac {3}{128} \, x \cos \left (4 \, x\right ) - \frac {15}{64} \, x \cos \left (2 \, x\right ) - \frac {1}{3456} \, {\left (18 \, x^{2} - 1\right )} \sin \left (6 \, x\right ) + \frac {3}{512} \, {\left (8 \, x^{2} - 1\right )} \sin \left (4 \, x\right ) - \frac {15}{128} \, {\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]
5/48*x^3 - 1/576*x*cos(6*x) + 3/128*x*cos(4*x) - 15/64*x*cos(2*x) - 1/3456 *(18*x^2 - 1)*sin(6*x) + 3/512*(8*x^2 - 1)*sin(4*x) - 15/128*(2*x^2 - 1)*s in(2*x)
Time = 0.42 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int x^2 \sin ^6(x) \, dx=\frac {15\,\sin \left (2\,x\right )}{128}-\frac {3\,\sin \left (4\,x\right )}{512}+\frac {\sin \left (6\,x\right )}{3456}-\frac {3\,x\,\left (2\,{\sin \left (2\,x\right )}^2-1\right )}{128}+\frac {x\,\left (2\,{\sin \left (3\,x\right )}^2-1\right )}{576}-\frac {15\,x^2\,\sin \left (2\,x\right )}{64}+\frac {3\,x^2\,\sin \left (4\,x\right )}{64}-\frac {x^2\,\sin \left (6\,x\right )}{192}+\frac {5\,x^3}{48}+\frac {15\,x\,\left (2\,{\sin \left (x\right )}^2-1\right )}{64} \]