Integrand size = 8, antiderivative size = 73 \[ \int x^3 \sin ^3(x) \, dx=\frac {40}{9} x \cos (x)-\frac {2}{3} x^3 \cos (x)-\frac {40 \sin (x)}{9}+2 x^2 \sin (x)+\frac {2}{9} x \cos (x) \sin ^2(x)-\frac {1}{3} x^3 \cos (x) \sin ^2(x)-\frac {2 \sin ^3(x)}{27}+\frac {1}{3} x^2 \sin ^3(x) \] Output:
40/9*x*cos(x)-2/3*x^3*cos(x)-40/9*sin(x)+2*x^2*sin(x)+2/9*x*cos(x)*sin(x)^ 2-1/3*x^3*cos(x)*sin(x)^2-2/27*sin(x)^3+1/3*x^2*sin(x)^3
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int x^3 \sin ^3(x) \, dx=\frac {1}{108} \left (-81 x \left (-6+x^2\right ) \cos (x)+3 x \left (-2+3 x^2\right ) \cos (3 x)+243 \left (-2+x^2\right ) \sin (x)-\left (-2+9 x^2\right ) \sin (3 x)\right ) \] Input:
Integrate[x^3*Sin[x]^3,x]
Output:
(-81*x*(-6 + x^2)*Cos[x] + 3*x*(-2 + 3*x^2)*Cos[3*x] + 243*(-2 + x^2)*Sin[ x] - (-2 + 9*x^2)*Sin[3*x])/108
Time = 0.65 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.26, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.000, Rules used = {3042, 3792, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117, 3791, 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.
| \(\displaystyle \int x^3 \sin ^3(x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \sin (x)^3dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {2}{3} \int x^3 \sin (x)dx-\frac {2}{3} \int x \sin ^3(x)dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \int x^3 \sin (x)dx-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {2}{3} \left (3 \int x^2 \cos (x)dx-x^3 \cos (x)\right )-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (3 \int x^2 \sin \left (x+\frac {\pi }{2}\right )dx-x^3 \cos (x)\right )-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {2}{3} \left (3 \left (2 \int -x \sin (x)dx+x^2 \sin (x)\right )-x^3 \cos (x)\right )-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} \left (3 \left (x^2 \sin (x)-2 \int x \sin (x)dx\right )-x^3 \cos (x)\right )-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (3 \left (x^2 \sin (x)-2 \int x \sin (x)dx\right )-x^3 \cos (x)\right )-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {2}{3} \left (3 \left (x^2 \sin (x)-2 (\int \cos (x)dx-x \cos (x))\right )-x^3 \cos (x)\right )-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (3 \left (x^2 \sin (x)-2 \left (\int \sin \left (x+\frac {\pi }{2}\right )dx-x \cos (x)\right )\right )-x^3 \cos (x)\right )-\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {2}{3} \int x \sin (x)^3dx-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)+\frac {2}{3} \left (3 \left (x^2 \sin (x)-2 (\sin (x)-x \cos (x))\right )-x^3 \cos (x)\right )\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {2}{3} \left (\frac {2}{3} \int x \sin (x)dx+\frac {\sin ^3(x)}{9}-\frac {1}{3} x \sin ^2(x) \cos (x)\right )-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)+\frac {2}{3} \left (3 \left (x^2 \sin (x)-2 (\sin (x)-x \cos (x))\right )-x^3 \cos (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2}{3} \left (\frac {2}{3} \int x \sin (x)dx+\frac {\sin ^3(x)}{9}-\frac {1}{3} x \sin ^2(x) \cos (x)\right )-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)+\frac {2}{3} \left (3 \left (x^2 \sin (x)-2 (\sin (x)-x \cos (x))\right )-x^3 \cos (x)\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {2}{3} \left (\frac {2}{3} (\int \cos (x)dx-x \cos (x))+\frac {\sin ^3(x)}{9}-\frac {1}{3} x \sin ^2(x) \cos (x)\right )-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)+\frac {2}{3} \left (3 \left (x^2 \sin (x)-2 (\sin (x)-x \cos (x))\right )-x^3 \cos (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2}{3} \left (\frac {2}{3} \left (\int \sin \left (x+\frac {\pi }{2}\right )dx-x \cos (x)\right )+\frac {\sin ^3(x)}{9}-\frac {1}{3} x \sin ^2(x) \cos (x)\right )-\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)+\frac {2}{3} \left (3 \left (x^2 \sin (x)-2 (\sin (x)-x \cos (x))\right )-x^3 \cos (x)\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {1}{3} x^3 \sin ^2(x) \cos (x)+\frac {1}{3} x^2 \sin ^3(x)+\frac {2}{3} \left (3 \left (x^2 \sin (x)-2 (\sin (x)-x \cos (x))\right )-x^3 \cos (x)\right )-\frac {2}{3} \left (\frac {\sin ^3(x)}{9}-\frac {1}{3} x \sin ^2(x) \cos (x)+\frac {2}{3} (\sin (x)-x \cos (x))\right )\) |
Input:
Int[x^3*Sin[x]^3,x]
Output:
-1/3*(x^3*Cos[x]*Sin[x]^2) + (x^2*Sin[x]^3)/3 - (2*(-1/3*(x*Cos[x]*Sin[x]^ 2) + Sin[x]^3/9 + (2*(-(x*Cos[x]) + Sin[x]))/3))/3 + (2*(-(x^3*Cos[x]) + 3 *(x^2*Sin[x] - 2*(-(x*Cos[x]) + Sin[x]))))/3
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_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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 = 0.75 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68
| method | result | size |
| risch | \(\left (-\frac {3}{4} x^{3}+\frac {9}{2} x \right ) \cos \left (x \right )+\frac {9 \left (x^{2}-2\right ) \sin \left (x \right )}{4}+\left (\frac {1}{12} x^{3}-\frac {1}{18} x \right ) \cos \left (3 x \right )-\frac {\left (9 x^{2}-2\right ) \sin \left (3 x \right )}{108}\) | \(50\) |
| default | \(-\frac {x^{3} \left (2+\sin \left (x \right )^{2}\right ) \cos \left (x \right )}{3}+2 x^{2} \sin \left (x \right )-\frac {40 \sin \left (x \right )}{9}+4 x \cos \left (x \right )+\frac {x^{2} \sin \left (x \right )^{3}}{3}+\frac {2 x \left (2+\sin \left (x \right )^{2}\right ) \cos \left (x \right )}{9}-\frac {2 \sin \left (x \right )^{3}}{27}\) | \(57\) |
| norman | \(\frac {\frac {40 x}{9}-\frac {2 x^{3}}{3}-\frac {496 \tan \left (\frac {x}{2}\right )^{3}}{27}-\frac {80 \tan \left (\frac {x}{2}\right )^{5}}{9}+\frac {16 x \tan \left (\frac {x}{2}\right )^{2}}{3}-\frac {16 x \tan \left (\frac {x}{2}\right )^{4}}{3}-\frac {40 x \tan \left (\frac {x}{2}\right )^{6}}{9}+4 x^{2} \tan \left (\frac {x}{2}\right )+\frac {32 x^{2} \tan \left (\frac {x}{2}\right )^{3}}{3}-2 x^{3} \tan \left (\frac {x}{2}\right )^{2}+2 x^{3} \tan \left (\frac {x}{2}\right )^{4}+\frac {2 x^{3} \tan \left (\frac {x}{2}\right )^{6}}{3}+4 \tan \left (\frac {x}{2}\right )^{5} x^{2}-\frac {80 \tan \left (\frac {x}{2}\right )}{9}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}\) | \(134\) |
| orering | \(\frac {20 \left (9 x^{4}-22 x^{2}-72\right ) \sin \left (x \right )^{3}}{27 x^{2}}-\frac {10 \left (3 x^{4}-2 x^{2}-84\right ) \left (3 x^{2} \sin \left (x \right )^{3}+3 x^{3} \cos \left (x \right ) \sin \left (x \right )^{2}\right )}{27 x^{4}}+\frac {4 \left (9 x^{2}-50\right ) \left (6 x \sin \left (x \right )^{3}+18 x^{2} \cos \left (x \right ) \sin \left (x \right )^{2}-3 x^{3} \sin \left (x \right )^{3}+6 x^{3} \cos \left (x \right )^{2} \sin \left (x \right )\right )}{27 x^{3}}-\frac {\left (3 x^{2}-20\right ) \left (6 \sin \left (x \right )^{3}+54 x \cos \left (x \right ) \sin \left (x \right )^{2}-27 x^{2} \sin \left (x \right )^{3}+54 x^{2} \cos \left (x \right )^{2} \sin \left (x \right )-21 x^{3} \cos \left (x \right ) \sin \left (x \right )^{2}+6 x^{3} \cos \left (x \right )^{3}\right )}{27 x^{2}}\) | \(180\) |
Input:
int(x^3*sin(x)^3,x,method=_RETURNVERBOSE)
Output:
(-3/4*x^3+9/2*x)*cos(x)+9/4*(x^2-2)*sin(x)+(1/12*x^3-1/18*x)*cos(3*x)-1/10 8*(9*x^2-2)*sin(3*x)
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int x^3 \sin ^3(x) \, dx=\frac {1}{9} \, {\left (3 \, x^{3} - 2 \, x\right )} \cos \left (x\right )^{3} - \frac {1}{3} \, {\left (3 \, x^{3} - 14 \, x\right )} \cos \left (x\right ) - \frac {1}{27} \, {\left ({\left (9 \, x^{2} - 2\right )} \cos \left (x\right )^{2} - 63 \, x^{2} + 122\right )} \sin \left (x\right ) \] Input:
integrate(x^3*sin(x)^3,x, algorithm="fricas")
Output:
1/9*(3*x^3 - 2*x)*cos(x)^3 - 1/3*(3*x^3 - 14*x)*cos(x) - 1/27*((9*x^2 - 2) *cos(x)^2 - 63*x^2 + 122)*sin(x)
Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.26 \[ \int x^3 \sin ^3(x) \, dx=- x^{3} \sin ^{2}{\left (x \right )} \cos {\left (x \right )} - \frac {2 x^{3} \cos ^{3}{\left (x \right )}}{3} + \frac {7 x^{2} \sin ^{3}{\left (x \right )}}{3} + 2 x^{2} \sin {\left (x \right )} \cos ^{2}{\left (x \right )} + \frac {14 x \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{3} + \frac {40 x \cos ^{3}{\left (x \right )}}{9} - \frac {122 \sin ^{3}{\left (x \right )}}{27} - \frac {40 \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{9} \] Input:
integrate(x**3*sin(x)**3,x)
Output:
-x**3*sin(x)**2*cos(x) - 2*x**3*cos(x)**3/3 + 7*x**2*sin(x)**3/3 + 2*x**2* sin(x)*cos(x)**2 + 14*x*sin(x)**2*cos(x)/3 + 40*x*cos(x)**3/9 - 122*sin(x) **3/27 - 40*sin(x)*cos(x)**2/9
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int x^3 \sin ^3(x) \, dx=\frac {1}{36} \, {\left (3 \, x^{3} - 2 \, x\right )} \cos \left (3 \, x\right ) - \frac {3}{4} \, {\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) - \frac {1}{108} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {9}{4} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \] Input:
integrate(x^3*sin(x)^3,x, algorithm="maxima")
Output:
1/36*(3*x^3 - 2*x)*cos(3*x) - 3/4*(x^3 - 6*x)*cos(x) - 1/108*(9*x^2 - 2)*s in(3*x) + 9/4*(x^2 - 2)*sin(x)
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int x^3 \sin ^3(x) \, dx=\frac {1}{36} \, {\left (3 \, x^{3} - 2 \, x\right )} \cos \left (3 \, x\right ) - \frac {3}{4} \, {\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) - \frac {1}{108} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {9}{4} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \] Input:
integrate(x^3*sin(x)^3,x, algorithm="giac")
Output:
1/36*(3*x^3 - 2*x)*cos(3*x) - 3/4*(x^3 - 6*x)*cos(x) - 1/108*(9*x^2 - 2)*s in(3*x) + 9/4*(x^2 - 2)*sin(x)
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int x^3 \sin ^3(x) \, dx=\frac {7\,x^2\,\sin \left (x\right )}{3}-\frac {2\,x\,{\cos \left (x\right )}^3}{9}-x^3\,\cos \left (x\right )-\frac {122\,\sin \left (x\right )}{27}+\frac {x^3\,{\cos \left (x\right )}^3}{3}+\frac {2\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{27}+\frac {14\,x\,\cos \left (x\right )}{3}-\frac {x^2\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{3} \] Input:
int(x^3*sin(x)^3,x)
Output:
(7*x^2*sin(x))/3 - (2*x*cos(x)^3)/9 - x^3*cos(x) - (122*sin(x))/27 + (x^3* cos(x)^3)/3 + (2*cos(x)^2*sin(x))/27 + (14*x*cos(x))/3 - (x^2*cos(x)^2*sin (x))/3
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int x^3 \sin ^3(x) \, dx=-\frac {\cos \left (x \right ) \sin \left (x \right )^{2} x^{3}}{3}+\frac {2 \cos \left (x \right ) \sin \left (x \right )^{2} x}{9}-\frac {2 \cos \left (x \right ) x^{3}}{3}+\frac {40 \cos \left (x \right ) x}{9}+\frac {\sin \left (x \right )^{3} x^{2}}{3}-\frac {2 \sin \left (x \right )^{3}}{27}+2 \sin \left (x \right ) x^{2}-\frac {40 \sin \left (x \right )}{9} \] Input:
int(x^3*sin(x)^3,x)
Output:
( - 9*cos(x)*sin(x)**2*x**3 + 6*cos(x)*sin(x)**2*x - 18*cos(x)*x**3 + 120* cos(x)*x + 9*sin(x)**3*x**2 - 2*sin(x)**3 + 54*sin(x)*x**2 - 120*sin(x))/2 7