3.52 \(\int \frac {\sin (\frac {1}{x})}{x^2} \, dx\)

Optimal. Leaf size=4 \[ \cos \left (\frac {1}{x}\right ) \]

[Out]

cos(1/x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3379, 2638} \[ \cos \left (\frac {1}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x^(-1)]/x^2,x]

[Out]

Cos[x^(-1)]

Rule 2638

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

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]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ \cos \left (\frac {1}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x^(-1)]/x^2,x]

[Out]

Cos[x^(-1)]

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 4, normalized size = 1.00 \[ \cos \left (\frac {1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/x)/x^2,x, algorithm="fricas")

[Out]

cos(1/x)

________________________________________________________________________________________

giac [A]  time = 0.01, size = 4, normalized size = 1.00 \[ \cos \left (\frac {1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/x)/x^2,x, algorithm="giac")

[Out]

cos(1/x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 5, normalized size = 1.25 \[ \cos \left (\frac {1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(1/x)/x^2,x)

[Out]

cos(1/x)

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 4, normalized size = 1.00 \[ \cos \left (\frac {1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/x)/x^2,x, algorithm="maxima")

[Out]

cos(1/x)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 4, normalized size = 1.00 \[ \cos \left (\frac {1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(1/x)/x^2,x)

[Out]

cos(1/x)

________________________________________________________________________________________

sympy [A]  time = 0.65, size = 3, normalized size = 0.75 \[ \cos {\left (\frac {1}{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/x)/x**2,x)

[Out]

cos(1/x)

________________________________________________________________________________________