3.776 \(\int x^{14} \sin (x^3) \, dx\)

Optimal. Leaf size=47 \[ \frac{4}{3} x^9 \sin \left (x^3\right )-8 x^3 \sin \left (x^3\right )-\frac{1}{3} x^{12} \cos \left (x^3\right )+4 x^6 \cos \left (x^3\right )-8 \cos \left (x^3\right ) \]

[Out]

-8*Cos[x^3] + 4*x^6*Cos[x^3] - (x^12*Cos[x^3])/3 - 8*x^3*Sin[x^3] + (4*x^9*Sin[x^3])/3

________________________________________________________________________________________

Rubi [A]  time = 0.0641899, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3379, 3296, 2638} \[ \frac{4}{3} x^9 \sin \left (x^3\right )-8 x^3 \sin \left (x^3\right )-\frac{1}{3} x^{12} \cos \left (x^3\right )+4 x^6 \cos \left (x^3\right )-8 \cos \left (x^3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^14*Sin[x^3],x]

[Out]

-8*Cos[x^3] + 4*x^6*Cos[x^3] - (x^12*Cos[x^3])/3 - 8*x^3*Sin[x^3] + (4*x^9*Sin[x^3])/3

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^{14} \sin \left (x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^4 \sin (x) \, dx,x,x^3\right )\\ &=-\frac{1}{3} x^{12} \cos \left (x^3\right )+\frac{4}{3} \operatorname{Subst}\left (\int x^3 \cos (x) \, dx,x,x^3\right )\\ &=-\frac{1}{3} x^{12} \cos \left (x^3\right )+\frac{4}{3} x^9 \sin \left (x^3\right )-4 \operatorname{Subst}\left (\int x^2 \sin (x) \, dx,x,x^3\right )\\ &=4 x^6 \cos \left (x^3\right )-\frac{1}{3} x^{12} \cos \left (x^3\right )+\frac{4}{3} x^9 \sin \left (x^3\right )-8 \operatorname{Subst}\left (\int x \cos (x) \, dx,x,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 )+8 \operatorname{Subst}\left (\int \sin (x) \, dx,x,x^3\right )\\ &=-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 )\\ \end{align*}

Mathematica [A]  time = 0.0315941, size = 35, normalized size = 0.74 \[ \frac{4}{3} x^3 \left (x^6-6\right ) \sin \left (x^3\right )-\frac{1}{3} \left (x^{12}-12 x^6+24\right ) \cos \left (x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^14*Sin[x^3],x]

[Out]

-((24 - 12*x^6 + x^12)*Cos[x^3])/3 + (4*x^3*(-6 + x^6)*Sin[x^3])/3

________________________________________________________________________________________

Maple [C]  time = 0.023, size = 64, normalized size = 1.4 \begin{align*} -{\frac{ \left ({x}^{12}+4\,i{x}^{9}-12\,{x}^{6}-24\,i{x}^{3}+24 \right ){{\rm e}^{i{x}^{3}}}}{6}}-{\frac{ \left ({x}^{12}-4\,i{x}^{9}-12\,{x}^{6}+24\,i{x}^{3}+24 \right ){{\rm e}^{-i{x}^{3}}}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14*sin(x^3),x)

[Out]

-1/6*(x^12+4*I*x^9-12*x^6-24*I*x^3+24)*exp(I*x^3)-1/6*(x^12-4*I*x^9-12*x^6+24*I*x^3+24)*exp(-I*x^3)

________________________________________________________________________________________

Maxima [A]  time = 0.963759, size = 43, normalized size = 0.91 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*sin(x^3),x, algorithm="maxima")

[Out]

-1/3*(x^12 - 12*x^6 + 24)*cos(x^3) + 4/3*(x^9 - 6*x^3)*sin(x^3)

________________________________________________________________________________________

Fricas [A]  time = 2.06486, size = 88, normalized size = 1.87 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*sin(x^3),x, algorithm="fricas")

[Out]

-1/3*(x^12 - 12*x^6 + 24)*cos(x^3) + 4/3*(x^9 - 6*x^3)*sin(x^3)

________________________________________________________________________________________

Sympy [A]  time = 123.675, size = 48, normalized size = 1.02 \begin{align*} - \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14*sin(x**3),x)

[Out]

-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)

________________________________________________________________________________________

Giac [A]  time = 1.08185, size = 43, normalized size = 0.91 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*sin(x^3),x, algorithm="giac")

[Out]

-1/3*(x^12 - 12*x^6 + 24)*cos(x^3) + 4/3*(x^9 - 6*x^3)*sin(x^3)