3.907 \(\int \frac{\cos (\frac{1}{x})}{x^5} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\sin \left (\frac{1}{x}\right )}{x^3}-\frac{3 \cos \left (\frac{1}{x}\right )}{x^2}+\frac{6 \sin \left (\frac{1}{x}\right )}{x}+6 \cos \left (\frac{1}{x}\right ) \]

[Out]

6*Cos[x^(-1)] - (3*Cos[x^(-1)])/x^2 - Sin[x^(-1)]/x^3 + (6*Sin[x^(-1)])/x

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Rubi [A]  time = 0.0485441, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3380, 3296, 2638} \[ -\frac{\sin \left (\frac{1}{x}\right )}{x^3}-\frac{3 \cos \left (\frac{1}{x}\right )}{x^2}+\frac{6 \sin \left (\frac{1}{x}\right )}{x}+6 \cos \left (\frac{1}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x^(-1)]/x^5,x]

[Out]

6*Cos[x^(-1)] - (3*Cos[x^(-1)])/x^2 - Sin[x^(-1)]/x^3 + (6*Sin[x^(-1)])/x

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]))

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 \frac{\cos \left (\frac{1}{x}\right )}{x^5} \, dx &=-\operatorname{Subst}\left (\int x^3 \cos (x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sin \left (\frac{1}{x}\right )}{x^3}+3 \operatorname{Subst}\left (\int x^2 \sin (x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 \cos \left (\frac{1}{x}\right )}{x^2}-\frac{\sin \left (\frac{1}{x}\right )}{x^3}+6 \operatorname{Subst}\left (\int x \cos (x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 \cos \left (\frac{1}{x}\right )}{x^2}-\frac{\sin \left (\frac{1}{x}\right )}{x^3}+\frac{6 \sin \left (\frac{1}{x}\right )}{x}-6 \operatorname{Subst}\left (\int \sin (x) \, dx,x,\frac{1}{x}\right )\\ &=6 \cos \left (\frac{1}{x}\right )-\frac{3 \cos \left (\frac{1}{x}\right )}{x^2}-\frac{\sin \left (\frac{1}{x}\right )}{x^3}+\frac{6 \sin \left (\frac{1}{x}\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.0252488, size = 32, normalized size = 0.94 \[ \frac{\left (6 x^2-1\right ) \sin \left (\frac{1}{x}\right )}{x^3}+\frac{3 \left (2 x^2-1\right ) \cos \left (\frac{1}{x}\right )}{x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x^(-1)]/x^5,x]

[Out]

(3*(-1 + 2*x^2)*Cos[x^(-1)])/x^2 + ((-1 + 6*x^2)*Sin[x^(-1)])/x^3

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Maple [A]  time = 0.008, size = 35, normalized size = 1. \begin{align*} 6\,\cos \left ({x}^{-1} \right ) -3\,{\frac{\cos \left ({x}^{-1} \right ) }{{x}^{2}}}-{\frac{\sin \left ({x}^{-1} \right ) }{{x}^{3}}}+6\,{\frac{\sin \left ({x}^{-1} \right ) }{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/x)/x^5,x)

[Out]

6*cos(1/x)-3*cos(1/x)/x^2-sin(1/x)/x^3+6*sin(1/x)/x

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Maxima [C]  time = 1.07376, size = 26, normalized size = 0.76 \begin{align*} \frac{1}{2} \, \Gamma \left (4, \frac{i}{x}\right ) + \frac{1}{2} \, \Gamma \left (4, -\frac{i}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/x)/x^5,x, algorithm="maxima")

[Out]

1/2*gamma(4, I/x) + 1/2*gamma(4, -I/x)

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Fricas [A]  time = 2.02326, size = 72, normalized size = 2.12 \begin{align*} \frac{3 \,{\left (2 \, x^{3} - x\right )} \cos \left (\frac{1}{x}\right ) +{\left (6 \, x^{2} - 1\right )} \sin \left (\frac{1}{x}\right )}{x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/x)/x^5,x, algorithm="fricas")

[Out]

(3*(2*x^3 - x)*cos(1/x) + (6*x^2 - 1)*sin(1/x))/x^3

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Sympy [A]  time = 4.17184, size = 32, normalized size = 0.94 \begin{align*} 6 \cos{\left (\frac{1}{x} \right )} + \frac{6 \sin{\left (\frac{1}{x} \right )}}{x} - \frac{3 \cos{\left (\frac{1}{x} \right )}}{x^{2}} - \frac{\sin{\left (\frac{1}{x} \right )}}{x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/x)/x**5,x)

[Out]

6*cos(1/x) + 6*sin(1/x)/x - 3*cos(1/x)/x**2 - sin(1/x)/x**3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{x}\right )}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/x)/x^5,x, algorithm="giac")

[Out]

integrate(cos(1/x)/x^5, x)