Integrand size = 12, antiderivative size = 65 \[ \int (2+3 x)^2 \sin ^3(x) \, dx=14 \cos (x)-\frac {2}{3} (2+3 x)^2 \cos (x)-\frac {2 \cos ^3(x)}{3}+4 (2+3 x) \sin (x)-\frac {1}{3} (2+3 x)^2 \cos (x) \sin ^2(x)+\frac {2}{3} (2+3 x) \sin ^3(x) \]
14*cos(x)-2/3*(2+3*x)^2*cos(x)-2/3*cos(x)^3+4*(2+3*x)*sin(x)-1/3*(2+3*x)^2 *cos(x)*sin(x)^2+2/3*(2+3*x)*sin(x)^3
Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int (2+3 x)^2 \sin ^3(x) \, dx=\frac {1}{12} \left (-9 \left (-14+12 x+9 x^2\right ) \cos (x)+\left (2+12 x+9 x^2\right ) \cos (3 x)-2 (2+3 x) (-27 \sin (x)+\sin (3 x))\right ) \]
(-9*(-14 + 12*x + 9*x^2)*Cos[x] + (2 + 12*x + 9*x^2)*Cos[3*x] - 2*(2 + 3*x )*(-27*Sin[x] + Sin[3*x]))/12
Time = 0.47 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3792, 3042, 3113, 2009, 3777, 3042, 3777, 25, 3042, 3118}
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.
\(\displaystyle \int (3 x+2)^2 \sin ^3(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (3 x+2)^2 \sin (x)^3dx\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle -2 \int \sin ^3(x)dx+\frac {2}{3} \int (3 x+2)^2 \sin (x)dx+\frac {2}{3} (3 x+2) \sin ^3(x)-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \int (3 x+2)^2 \sin (x)dx-2 \int \sin (x)^3dx+\frac {2}{3} (3 x+2) \sin ^3(x)-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle \frac {2}{3} \int (3 x+2)^2 \sin (x)dx+2 \int \left (1-\cos ^2(x)\right )d\cos (x)+\frac {2}{3} (3 x+2) \sin ^3(x)-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \int (3 x+2)^2 \sin (x)dx+\frac {2}{3} (3 x+2) \sin ^3(x)+2 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {2}{3} \left (6 \int (3 x+2) \cos (x)dx-(3 x+2)^2 \cos (x)\right )+\frac {2}{3} (3 x+2) \sin ^3(x)+2 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \left (6 \int (3 x+2) \sin \left (x+\frac {\pi }{2}\right )dx-(3 x+2)^2 \cos (x)\right )+\frac {2}{3} (3 x+2) \sin ^3(x)+2 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {2}{3} \left (6 (3 \int -\sin (x)dx+(3 x+2) \sin (x))-(3 x+2)^2 \cos (x)\right )+\frac {2}{3} (3 x+2) \sin ^3(x)+2 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} \left (6 ((3 x+2) \sin (x)-3 \int \sin (x)dx)-(3 x+2)^2 \cos (x)\right )+\frac {2}{3} (3 x+2) \sin ^3(x)+2 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \left (6 ((3 x+2) \sin (x)-3 \int \sin (x)dx)-(3 x+2)^2 \cos (x)\right )+\frac {2}{3} (3 x+2) \sin ^3(x)+2 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {2}{3} (3 x+2) \sin ^3(x)+2 \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)+\frac {2}{3} \left (6 ((3 x+2) \sin (x)+3 \cos (x))-(3 x+2)^2 \cos (x)\right )\) |
2*(Cos[x] - Cos[x]^3/3) - ((2 + 3*x)^2*Cos[x]*Sin[x]^2)/3 + (2*(2 + 3*x)*S in[x]^3)/3 + (2*(-((2 + 3*x)^2*Cos[x]) + 6*(3*Cos[x] + (2 + 3*x)*Sin[x]))) /3
3.10.16.3.1 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[((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 = 2.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\left (-\frac {27}{4} x^{2}-9 x +\frac {21}{2}\right ) \cos \left (x \right )+\frac {9 \left (2+3 x \right ) \sin \left (x \right )}{2}+\left (\frac {3}{4} x^{2}+x +\frac {1}{6}\right ) \cos \left (3 x \right )-\frac {\left (2+3 x \right ) \sin \left (3 x \right )}{6}\) | \(48\) |
default | \(-3 x^{2} \left (2+\sin \left (x \right )^{2}\right ) \cos \left (x \right )+12 \cos \left (x \right )+12 x \sin \left (x \right )+2 \sin \left (x \right )^{3} x -\frac {2 \left (2+\sin \left (x \right )^{2}\right ) \cos \left (x \right )}{3}-4 x \left (2+\sin \left (x \right )^{2}\right ) \cos \left (x \right )+\frac {4 \sin \left (x \right )^{3}}{3}+8 \sin \left (x \right )\) | \(62\) |
norman | \(\frac {24 \tan \left (\frac {x}{2}\right )^{4}+40 \tan \left (\frac {x}{2}\right )^{2}-8 x -6 x^{2}+\frac {128 \tan \left (\frac {x}{2}\right )^{3}}{3}+16 \tan \left (\frac {x}{2}\right )^{5}+24 x \tan \left (\frac {x}{2}\right )-24 x \tan \left (\frac {x}{2}\right )^{2}+64 x \tan \left (\frac {x}{2}\right )^{3}+24 x \tan \left (\frac {x}{2}\right )^{4}+24 x \tan \left (\frac {x}{2}\right )^{5}+8 x \tan \left (\frac {x}{2}\right )^{6}-18 x^{2} \tan \left (\frac {x}{2}\right )^{2}+18 x^{2} \tan \left (\frac {x}{2}\right )^{4}+6 x^{2} \tan \left (\frac {x}{2}\right )^{6}+16 \tan \left (\frac {x}{2}\right )+\frac {64}{3}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}\) | \(145\) |
(-27/4*x^2-9*x+21/2)*cos(x)+9/2*(2+3*x)*sin(x)+(3/4*x^2+x+1/6)*cos(3*x)-1/ 6*(2+3*x)*sin(3*x)
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int (2+3 x)^2 \sin ^3(x) \, dx=\frac {1}{3} \, {\left (9 \, x^{2} + 12 \, x + 2\right )} \cos \left (x\right )^{3} - {\left (9 \, x^{2} + 12 \, x - 10\right )} \cos \left (x\right ) - \frac {2}{3} \, {\left ({\left (3 \, x + 2\right )} \cos \left (x\right )^{2} - 21 \, x - 14\right )} \sin \left (x\right ) \]
1/3*(9*x^2 + 12*x + 2)*cos(x)^3 - (9*x^2 + 12*x - 10)*cos(x) - 2/3*((3*x + 2)*cos(x)^2 - 21*x - 14)*sin(x)
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54 \[ \int (2+3 x)^2 \sin ^3(x) \, dx=- 9 x^{2} \sin ^{2}{\left (x \right )} \cos {\left (x \right )} - 6 x^{2} \cos ^{3}{\left (x \right )} + 14 x \sin ^{3}{\left (x \right )} - 12 x \sin ^{2}{\left (x \right )} \cos {\left (x \right )} + 12 x \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - 8 x \cos ^{3}{\left (x \right )} + \frac {28 \sin ^{3}{\left (x \right )}}{3} + 10 \sin ^{2}{\left (x \right )} \cos {\left (x \right )} + 8 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} + \frac {32 \cos ^{3}{\left (x \right )}}{3} \]
-9*x**2*sin(x)**2*cos(x) - 6*x**2*cos(x)**3 + 14*x*sin(x)**3 - 12*x*sin(x) **2*cos(x) + 12*x*sin(x)*cos(x)**2 - 8*x*cos(x)**3 + 28*sin(x)**3/3 + 10*s in(x)**2*cos(x) + 8*sin(x)*cos(x)**2 + 32*cos(x)**3/3
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int (2+3 x)^2 \sin ^3(x) \, dx=\frac {4}{3} \, \cos \left (x\right )^{3} + \frac {1}{12} \, {\left (9 \, x^{2} - 2\right )} \cos \left (3 \, x\right ) + x \cos \left (3 \, x\right ) - \frac {27}{4} \, {\left (x^{2} - 2\right )} \cos \left (x\right ) - 9 \, x \cos \left (x\right ) - \frac {1}{2} \, x \sin \left (3 \, x\right ) + \frac {27}{2} \, x \sin \left (x\right ) - 4 \, \cos \left (x\right ) - \frac {1}{3} \, \sin \left (3 \, x\right ) + 9 \, \sin \left (x\right ) \]
4/3*cos(x)^3 + 1/12*(9*x^2 - 2)*cos(3*x) + x*cos(3*x) - 27/4*(x^2 - 2)*cos (x) - 9*x*cos(x) - 1/2*x*sin(3*x) + 27/2*x*sin(x) - 4*cos(x) - 1/3*sin(3*x ) + 9*sin(x)
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int (2+3 x)^2 \sin ^3(x) \, dx=\frac {1}{12} \, {\left (9 \, x^{2} + 12 \, x + 2\right )} \cos \left (3 \, x\right ) - \frac {3}{4} \, {\left (9 \, x^{2} + 12 \, x - 14\right )} \cos \left (x\right ) - \frac {1}{6} \, {\left (3 \, x + 2\right )} \sin \left (3 \, x\right ) + \frac {9}{2} \, {\left (3 \, x + 2\right )} \sin \left (x\right ) \]
1/12*(9*x^2 + 12*x + 2)*cos(3*x) - 3/4*(9*x^2 + 12*x - 14)*cos(x) - 1/6*(3 *x + 2)*sin(3*x) + 9/2*(3*x + 2)*sin(x)
Time = 26.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int (2+3 x)^2 \sin ^3(x) \, dx=10\,\cos \left (x\right )+\frac {28\,\sin \left (x\right )}{3}-9\,x^2\,\cos \left (x\right )+4\,x\,{\cos \left (x\right )}^3+\frac {2\,{\cos \left (x\right )}^3}{3}+3\,x^2\,{\cos \left (x\right )}^3-\frac {4\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{3}-12\,x\,\cos \left (x\right )+14\,x\,\sin \left (x\right )-2\,x\,{\cos \left (x\right )}^2\,\sin \left (x\right ) \]