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]

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

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Rubi [A]  time = 0.0123809, antiderivative size = 10, normalized size of antiderivative = 1., 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 = 0.0053399, size = 10, normalized size = 1. \[ \frac{1}{2} \cos ^2\left (\frac{1}{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 9, normalized size = 0.9 \begin{align*}{\frac{ \left ( \cos \left ({x}^{-1} \right ) \right ) ^{2}}{2}} \end{align*}

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

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Maxima [A]  time = 0.949932, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{2} \, \cos \left (\frac{1}{x}\right )^{2} \end{align*}

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

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Fricas [A]  time = 2.19054, size = 22, normalized size = 2.2 \begin{align*} \frac{1}{2} \, \cos \left (\frac{1}{x}\right )^{2} \end{align*}

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

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Sympy [B]  time = 1.74315, size = 31, normalized size = 3.1 \begin{align*} - \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} \end{align*}

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)

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Giac [A]  time = 1.06668, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{2} \, \cos \left (\frac{1}{x}\right )^{2} \end{align*}

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