3.108 \(\int \frac{\sin (a+\frac{b}{x})}{x^3} \, dx\)

Optimal. Leaf size=29 \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b^2} \]

[Out]

Cos[a + b/x]/(b*x) - Sin[a + b/x]/b^2

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Rubi [A]  time = 0.0246476, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3379, 3296, 2637} \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/x]/x^3,x]

[Out]

Cos[a + b/x]/(b*x) - Sin[a + b/x]/b^2

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 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{\sin \left (a+\frac{b}{x}\right )}{x^3} \, dx &=-\operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x}-\frac{\operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\frac{1}{x}\right )}{b}\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.0040076, size = 29, normalized size = 1. \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/x]/x^3,x]

[Out]

Cos[a + b/x]/(b*x) - Sin[a + b/x]/b^2

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Maple [A]  time = 0.007, size = 42, normalized size = 1.5 \begin{align*} -{\frac{1}{{b}^{2}} \left ( \sin \left ( a+{\frac{b}{x}} \right ) - \left ( a+{\frac{b}{x}} \right ) \cos \left ( a+{\frac{b}{x}} \right ) +a\cos \left ( a+{\frac{b}{x}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/x)/x^3,x)

[Out]

-1/b^2*(sin(a+b/x)-(a+b/x)*cos(a+b/x)+a*cos(a+b/x))

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Maxima [C]  time = 1.13393, size = 68, normalized size = 2.34 \begin{align*} -\frac{{\left (i \, \Gamma \left (2, \frac{i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) +{\left (\Gamma \left (2, \frac{i \, b}{x}\right ) + \Gamma \left (2, -\frac{i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x^3,x, algorithm="maxima")

[Out]

-1/2*((I*gamma(2, I*b/x) - I*gamma(2, -I*b/x))*cos(a) + (gamma(2, I*b/x) + gamma(2, -I*b/x))*sin(a))/b^2

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Fricas [A]  time = 1.56024, size = 69, normalized size = 2.38 \begin{align*} \frac{b \cos \left (\frac{a x + b}{x}\right ) - x \sin \left (\frac{a x + b}{x}\right )}{b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x^3,x, algorithm="fricas")

[Out]

(b*cos((a*x + b)/x) - x*sin((a*x + b)/x))/(b^2*x)

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Sympy [A]  time = 2.42722, size = 29, normalized size = 1. \begin{align*} \begin{cases} \frac{\cos{\left (a + \frac{b}{x} \right )}}{b x} - \frac{\sin{\left (a + \frac{b}{x} \right )}}{b^{2}} & \text{for}\: b \neq 0 \\- \frac{\sin{\left (a \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x**3,x)

[Out]

Piecewise((cos(a + b/x)/(b*x) - sin(a + b/x)/b**2, Ne(b, 0)), (-sin(a)/(2*x**2), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{x}\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x^3,x, algorithm="giac")

[Out]

integrate(sin(a + b/x)/x^3, x)