3.53 \(\int x^5 \cos (x^3) \, dx\)

Optimal. Leaf size=20 \[ \frac {1}{3} x^3 \sin \left (x^3\right )+\frac {\cos \left (x^3\right )}{3} \]

[Out]

1/3*cos(x^3)+1/3*x^3*sin(x^3)

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3380, 3296, 2638} \[ \frac {1}{3} x^3 \sin \left (x^3\right )+\frac {\cos \left (x^3\right )}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Cos[x^3],x]

[Out]

Cos[x^3]/3 + (x^3*Sin[x^3])/3

Rule 2638

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

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 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[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]))

Rubi steps

\begin {align*} \int x^5 \cos \left (x^3\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x \cos (x) \, dx,x,x^3\right )\\ &=\frac {1}{3} x^3 \sin \left (x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \sin (x) \, dx,x,x^3\right )\\ &=\frac {\cos \left (x^3\right )}{3}+\frac {1}{3} x^3 \sin \left (x^3\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \[ \frac {1}{3} x^3 \sin \left (x^3\right )+\frac {\cos \left (x^3\right )}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Cos[x^3],x]

[Out]

Cos[x^3]/3 + (x^3*Sin[x^3])/3

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fricas [A]  time = 0.44, size = 16, normalized size = 0.80 \[ \frac {1}{3} \, x^{3} \sin \left (x^{3}\right ) + \frac {1}{3} \, \cos \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cos(x^3),x, algorithm="fricas")

[Out]

1/3*x^3*sin(x^3) + 1/3*cos(x^3)

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giac [A]  time = 0.96, size = 16, normalized size = 0.80 \[ \frac {1}{3} \, x^{3} \sin \left (x^{3}\right ) + \frac {1}{3} \, \cos \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cos(x^3),x, algorithm="giac")

[Out]

1/3*x^3*sin(x^3) + 1/3*cos(x^3)

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maple [A]  time = 0.02, size = 17, normalized size = 0.85 \[ \frac {x^{3} \sin \left (x^{3}\right )}{3}+\frac {\cos \left (x^{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cos(x^3),x)

[Out]

1/3*cos(x^3)+1/3*x^3*sin(x^3)

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maxima [A]  time = 0.44, size = 16, normalized size = 0.80 \[ \frac {1}{3} \, x^{3} \sin \left (x^{3}\right ) + \frac {1}{3} \, \cos \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cos(x^3),x, algorithm="maxima")

[Out]

1/3*x^3*sin(x^3) + 1/3*cos(x^3)

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mupad [B]  time = 0.19, size = 16, normalized size = 0.80 \[ \frac {\cos \left (x^3\right )}{3}+\frac {x^3\,\sin \left (x^3\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cos(x^3),x)

[Out]

cos(x^3)/3 + (x^3*sin(x^3))/3

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sympy [A]  time = 1.91, size = 15, normalized size = 0.75 \[ \frac {x^{3} \sin {\left (x^{3} \right )}}{3} + \frac {\cos {\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*cos(x**3),x)

[Out]

x**3*sin(x**3)/3 + cos(x**3)/3

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