3.37 \(\int \frac{\cos (a+\frac{b}{x})}{x^2} \, dx\)

Optimal. Leaf size=13 \[ -\frac{\sin \left (a+\frac{b}{x}\right )}{b} \]

[Out]

-(Sin[a + b/x]/b)

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Rubi [A]  time = 0.0144733, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3380, 2637} \[ -\frac{\sin \left (a+\frac{b}{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b/x]/x^2,x]

[Out]

-(Sin[a + b/x]/b)

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0028499, size = 13, normalized size = 1. \[ -\frac{\sin \left (a+\frac{b}{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b/x]/x^2,x]

[Out]

-(Sin[a + b/x]/b)

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Maple [A]  time = 0.026, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{b}\sin \left ( a+{\frac{b}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b/x)/x^2,x)

[Out]

-sin(a+b/x)/b

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Maxima [A]  time = 1.055, size = 18, normalized size = 1.38 \begin{align*} -\frac{\sin \left (a + \frac{b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x)/x^2,x, algorithm="maxima")

[Out]

-sin(a + b/x)/b

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Fricas [A]  time = 1.51144, size = 28, normalized size = 2.15 \begin{align*} -\frac{\sin \left (\frac{a x + b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x)/x^2,x, algorithm="fricas")

[Out]

-sin((a*x + b)/x)/b

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Sympy [A]  time = 1.7594, size = 15, normalized size = 1.15 \begin{align*} \begin{cases} - \frac{\sin{\left (a + \frac{b}{x} \right )}}{b} & \text{for}\: b \neq 0 \\- \frac{\cos{\left (a \right )}}{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x)/x**2,x)

[Out]

Piecewise((-sin(a + b/x)/b, Ne(b, 0)), (-cos(a)/x, True))

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Giac [A]  time = 1.13324, size = 18, normalized size = 1.38 \begin{align*} -\frac{\sin \left (a + \frac{b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x)/x^2,x, algorithm="giac")

[Out]

-sin(a + b/x)/b