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

Optimal. Leaf size=10 \[ -\frac {1}{2} \sin ^2\left (\frac {1}{x}\right ) \]

[Out]

-1/2*sin(1/x)^2

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3441} \[ -\frac {1}{2} \sin ^2\left (\frac {1}{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3441

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[Sin[a + b*
x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \frac {1}{2} \cos ^2\left (\frac {1}{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[x^(-1)]^2/2

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 8, normalized size = 0.80 \[ \frac {1}{2} \, \cos \left (\frac {1}{x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cos(1/x)^2

________________________________________________________________________________________

giac [A]  time = 0.13, size = 8, normalized size = 0.80 \[ \frac {1}{2} \, \cos \left (\frac {1}{x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cos(1/x)^2

________________________________________________________________________________________

maple [A]  time = 0.00, size = 9, normalized size = 0.90 \[ \frac {\left (\cos ^{2}\left (\frac {1}{x}\right )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cos(1/x)^2

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 8, normalized size = 0.80 \[ \frac {1}{2} \, \cos \left (\frac {1}{x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cos(1/x)^2

________________________________________________________________________________________

mupad [B]  time = 2.92, size = 8, normalized size = 0.80 \[ \frac {{\cos \left (\frac {1}{x}\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(1/x)^2/2

________________________________________________________________________________________

sympy [B]  time = 1.28, size = 31, normalized size = 3.10 \[ - \frac {2 \tan ^{2}{\left (\frac {1}{2 x} \right )}}{\tan ^{4}{\left (\frac {1}{2 x} \right )} + 2 \tan ^{2}{\left (\frac {1}{2 x} \right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*tan(1/(2*x))**2/(tan(1/(2*x))**4 + 2*tan(1/(2*x))**2 + 1)

________________________________________________________________________________________