3.107 \(\int \frac{\sin (a+\frac{b}{x})}{x^2} \, dx\)

Optimal. Leaf size=12 \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b} \]

[Out]

Cos[a + b/x]/b

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Rubi [A]  time = 0.0138755, antiderivative size = 12, 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 = {3379, 2638} \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/x]/x^2,x]

[Out]

Cos[a + b/x]/b

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]))

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

Mathematica [A]  time = 0.01326, size = 12, normalized size = 1. \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/x]/x^2,x]

[Out]

Cos[a + b/x]/b

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Maple [A]  time = 0.004, size = 13, normalized size = 1.1 \begin{align*}{\frac{1}{b}\cos \left ( a+{\frac{b}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(a+b/x)/b

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Maxima [A]  time = 0.956441, size = 16, normalized size = 1.33 \begin{align*} \frac{\cos \left (a + \frac{b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(a + b/x)/b

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Fricas [A]  time = 1.69086, size = 27, normalized size = 2.25 \begin{align*} \frac{\cos \left (\frac{a x + b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos((a*x + b)/x)/b

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Sympy [A]  time = 1.2605, size = 14, normalized size = 1.17 \begin{align*} \begin{cases} \frac{\cos{\left (a + \frac{b}{x} \right )}}{b} & \text{for}\: b \neq 0 \\- \frac{\sin{\left (a \right )}}{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10533, size = 16, normalized size = 1.33 \begin{align*} \frac{\cos \left (a + \frac{b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(a + b/x)/b