Integrand size = 8, antiderivative size = 47 \[ \int x^{14} \sin \left (x^3\right ) \, dx=-8 \cos \left (x^3\right )+4 x^6 \cos \left (x^3\right )-\frac {1}{3} x^{12} \cos \left (x^3\right )-8 x^3 \sin \left (x^3\right )+\frac {4}{3} x^9 \sin \left (x^3\right ) \] Output:
-8*cos(x^3)+4*x^6*cos(x^3)-1/3*x^12*cos(x^3)-8*x^3*sin(x^3)+4/3*x^9*sin(x^ 3)
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int x^{14} \sin \left (x^3\right ) \, dx=-\frac {1}{3} \left (24-12 x^6+x^{12}\right ) \cos \left (x^3\right )+\frac {4}{3} x^3 \left (-6+x^6\right ) \sin \left (x^3\right ) \] Input:
Integrate[x^14*Sin[x^3],x]
Output:
-1/3*((24 - 12*x^6 + x^12)*Cos[x^3]) + (4*x^3*(-6 + x^6)*Sin[x^3])/3
Time = 0.45 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.625, Rules used = {3860, 3042, 3777, 3042, 3777, 25, 3042, 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.
\(\displaystyle \int x^{14} \sin \left (x^3\right ) \, dx\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle \frac {1}{3} \int x^{12} \sin \left (x^3\right )dx^3\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \int x^{12} \sin \left (x^3\right )dx^3\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} \left (4 \int x^9 \cos \left (x^3\right )dx^3-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (4 \int x^9 \sin \left (x^3+\frac {\pi }{2}\right )dx^3-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} \left (4 \left (3 \int -x^6 \sin \left (x^3\right )dx^3+x^9 \sin \left (x^3\right )\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \int x^6 \sin \left (x^3\right )dx^3\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \int x^6 \sin \left (x^3\right )dx^3\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \left (2 \int x^3 \cos \left (x^3\right )dx^3-x^6 \cos \left (x^3\right )\right )\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \left (2 \int x^3 \sin \left (x^3+\frac {\pi }{2}\right )dx^3-x^6 \cos \left (x^3\right )\right )\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \left (2 \left (\int -\sin \left (x^3\right )dx^3+x^3 \sin \left (x^3\right )\right )-x^6 \cos \left (x^3\right )\right )\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \left (2 \left (x^3 \sin \left (x^3\right )-\int \sin \left (x^3\right )dx^3\right )-x^6 \cos \left (x^3\right )\right )\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \left (2 \left (x^3 \sin \left (x^3\right )-\int \sin \left (x^3\right )dx^3\right )-x^6 \cos \left (x^3\right )\right )\right )-x^{12} \cos \left (x^3\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {1}{3} \left (4 \left (x^9 \sin \left (x^3\right )-3 \left (2 \left (x^3 \sin \left (x^3\right )+\cos \left (x^3\right )\right )-x^6 \cos \left (x^3\right )\right )\right )-x^{12} \cos \left (x^3\right )\right )\) |
Input:
Int[x^14*Sin[x^3],x]
Output:
(-(x^12*Cos[x^3]) + 4*(x^9*Sin[x^3] - 3*(-(x^6*Cos[x^3]) + 2*(Cos[x^3] + x ^3*Sin[x^3]))))/3
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[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\left (-\frac {1}{3} x^{12}+4 x^{6}-8\right ) \cos \left (x^{3}\right )+\frac {4 x^{3} \left (x^{6}-6\right ) \sin \left (x^{3}\right )}{3}\) | \(33\) |
parallelrisch | \(-8+\frac {\left (-x^{12}+12 x^{6}-24\right ) \cos \left (x^{3}\right )}{3}+\frac {4 \left (x^{9}-6 x^{3}\right ) \sin \left (x^{3}\right )}{3}\) | \(36\) |
meijerg | \(\frac {16 \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{12}-\frac {9}{2} x^{6}+9\right ) \cos \left (x^{3}\right )}{6 \sqrt {\pi }}-\frac {x^{3} \left (-\frac {3 x^{6}}{2}+9\right ) \sin \left (x^{3}\right )}{6 \sqrt {\pi }}\right )}{3}\) | \(52\) |
orering | \(\frac {2 \left (13 x^{12}-120 x^{6}+168\right ) \sin \left (x^{3}\right )}{9 x^{3}}-\frac {\left (x^{12}-12 x^{6}+24\right ) \left (14 x^{13} \sin \left (x^{3}\right )+3 x^{16} \cos \left (x^{3}\right )\right )}{9 x^{16}}\) | \(57\) |
Input:
int(x^14*sin(x^3),x,method=_RETURNVERBOSE)
Output:
(-1/3*x^12+4*x^6-8)*cos(x^3)+4/3*x^3*(x^6-6)*sin(x^3)
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.68 \[ \int x^{14} \sin \left (x^3\right ) \, dx=-\frac {1}{3} \, {\left (x^{12} - 12 \, x^{6} + 24\right )} \cos \left (x^{3}\right ) + \frac {4}{3} \, {\left (x^{9} - 6 \, x^{3}\right )} \sin \left (x^{3}\right ) \] Input:
integrate(x^14*sin(x^3),x, algorithm="fricas")
Output:
-1/3*(x^12 - 12*x^6 + 24)*cos(x^3) + 4/3*(x^9 - 6*x^3)*sin(x^3)
Time = 3.65 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int x^{14} \sin \left (x^3\right ) \, dx=- \frac {x^{12} \cos {\left (x^{3} \right )}}{3} + \frac {4 x^{9} \sin {\left (x^{3} \right )}}{3} + 4 x^{6} \cos {\left (x^{3} \right )} - 8 x^{3} \sin {\left (x^{3} \right )} - 8 \cos {\left (x^{3} \right )} \] Input:
integrate(x**14*sin(x**3),x)
Output:
-x**12*cos(x**3)/3 + 4*x**9*sin(x**3)/3 + 4*x**6*cos(x**3) - 8*x**3*sin(x* *3) - 8*cos(x**3)
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.68 \[ \int x^{14} \sin \left (x^3\right ) \, dx=-\frac {1}{3} \, {\left (x^{12} - 12 \, x^{6} + 24\right )} \cos \left (x^{3}\right ) + \frac {4}{3} \, {\left (x^{9} - 6 \, x^{3}\right )} \sin \left (x^{3}\right ) \] Input:
integrate(x^14*sin(x^3),x, algorithm="maxima")
Output:
-1/3*(x^12 - 12*x^6 + 24)*cos(x^3) + 4/3*(x^9 - 6*x^3)*sin(x^3)
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.68 \[ \int x^{14} \sin \left (x^3\right ) \, dx=-\frac {1}{3} \, {\left (x^{12} - 12 \, x^{6} + 24\right )} \cos \left (x^{3}\right ) + \frac {4}{3} \, {\left (x^{9} - 6 \, x^{3}\right )} \sin \left (x^{3}\right ) \] Input:
integrate(x^14*sin(x^3),x, algorithm="giac")
Output:
-1/3*(x^12 - 12*x^6 + 24)*cos(x^3) + 4/3*(x^9 - 6*x^3)*sin(x^3)
Time = 16.89 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int x^{14} \sin \left (x^3\right ) \, dx=4\,x^6\,\cos \left (x^3\right )-8\,\cos \left (x^3\right )-\frac {x^{12}\,\cos \left (x^3\right )}{3}-8\,x^3\,\sin \left (x^3\right )+\frac {4\,x^9\,\sin \left (x^3\right )}{3} \] Input:
int(x^14*sin(x^3),x)
Output:
4*x^6*cos(x^3) - 8*cos(x^3) - (x^12*cos(x^3))/3 - 8*x^3*sin(x^3) + (4*x^9* sin(x^3))/3
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int x^{14} \sin \left (x^3\right ) \, dx=-\frac {\cos \left (x^{3}\right ) x^{12}}{3}+4 \cos \left (x^{3}\right ) x^{6}-8 \cos \left (x^{3}\right )+\frac {4 \sin \left (x^{3}\right ) x^{9}}{3}-8 \sin \left (x^{3}\right ) x^{3} \] Input:
int(x^14*sin(x^3),x)
Output:
( - cos(x**3)*x**12 + 12*cos(x**3)*x**6 - 24*cos(x**3) + 4*sin(x**3)*x**9 - 24*sin(x**3)*x**3)/3