3.110 \(\int \frac{\sin (a+\frac{b}{x})}{x^5} \, dx\)

Optimal. Leaf size=61 \[ -\frac{3 \sin \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \sin \left (a+\frac{b}{x}\right )}{b^4}-\frac{6 \cos \left (a+\frac{b}{x}\right )}{b^3 x}+\frac{\cos \left (a+\frac{b}{x}\right )}{b x^3} \]

[Out]

Cos[a + b/x]/(b*x^3) - (6*Cos[a + b/x])/(b^3*x) + (6*Sin[a + b/x])/b^4 - (3*Sin[a + b/x])/(b^2*x^2)

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Rubi [A]  time = 0.0676861, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3379, 3296, 2637} \[ -\frac{3 \sin \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \sin \left (a+\frac{b}{x}\right )}{b^4}-\frac{6 \cos \left (a+\frac{b}{x}\right )}{b^3 x}+\frac{\cos \left (a+\frac{b}{x}\right )}{b x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/x]/x^5,x]

[Out]

Cos[a + b/x]/(b*x^3) - (6*Cos[a + b/x])/(b^3*x) + (6*Sin[a + b/x])/b^4 - (3*Sin[a + b/x])/(b^2*x^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^5} \, dx &=-\operatorname{Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x^3}-\frac{3 \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\frac{1}{x}\right )}{b}\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x^3}-\frac{3 \sin \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\frac{1}{x}\right )}{b^2}\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x^3}-\frac{6 \cos \left (a+\frac{b}{x}\right )}{b^3 x}-\frac{3 \sin \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\frac{1}{x}\right )}{b^3}\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x^3}-\frac{6 \cos \left (a+\frac{b}{x}\right )}{b^3 x}+\frac{6 \sin \left (a+\frac{b}{x}\right )}{b^4}-\frac{3 \sin \left (a+\frac{b}{x}\right )}{b^2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0045402, size = 61, normalized size = 1. \[ -\frac{3 \sin \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \sin \left (a+\frac{b}{x}\right )}{b^4}-\frac{6 \cos \left (a+\frac{b}{x}\right )}{b^3 x}+\frac{\cos \left (a+\frac{b}{x}\right )}{b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/x]/x^5,x]

[Out]

Cos[a + b/x]/(b*x^3) - (6*Cos[a + b/x])/(b^3*x) + (6*Sin[a + b/x])/b^4 - (3*Sin[a + b/x])/(b^2*x^2)

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Maple [B]  time = 0.008, size = 165, normalized size = 2.7 \begin{align*} -{\frac{1}{{b}^{4}} \left ( - \left ( a+{\frac{b}{x}} \right ) ^{3}\cos \left ( a+{\frac{b}{x}} \right ) +3\, \left ( a+{\frac{b}{x}} \right ) ^{2}\sin \left ( a+{\frac{b}{x}} \right ) -6\,\sin \left ( a+{\frac{b}{x}} \right ) +6\, \left ( a+{\frac{b}{x}} \right ) \cos \left ( a+{\frac{b}{x}} \right ) -3\,a \left ( - \left ( a+{\frac{b}{x}} \right ) ^{2}\cos \left ( a+{\frac{b}{x}} \right ) +2\,\cos \left ( a+{\frac{b}{x}} \right ) +2\, \left ( a+{\frac{b}{x}} \right ) \sin \left ( a+{\frac{b}{x}} \right ) \right ) +3\,{a}^{2} \left ( \sin \left ( a+{\frac{b}{x}} \right ) - \left ( a+{\frac{b}{x}} \right ) \cos \left ( a+{\frac{b}{x}} \right ) \right ) +{a}^{3}\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^5,x)

[Out]

-1/b^4*(-(a+b/x)^3*cos(a+b/x)+3*(a+b/x)^2*sin(a+b/x)-6*sin(a+b/x)+6*(a+b/x)*cos(a+b/x)-3*a*(-(a+b/x)^2*cos(a+b
/x)+2*cos(a+b/x)+2*(a+b/x)*sin(a+b/x))+3*a^2*(sin(a+b/x)-(a+b/x)*cos(a+b/x))+a^3*cos(a+b/x))

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Maxima [C]  time = 1.14778, size = 68, normalized size = 1.11 \begin{align*} \frac{{\left (i \, \Gamma \left (4, \frac{i \, b}{x}\right ) - i \, \Gamma \left (4, -\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) +{\left (\Gamma \left (4, \frac{i \, b}{x}\right ) + \Gamma \left (4, -\frac{i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x^5,x, algorithm="maxima")

[Out]

1/2*((I*gamma(4, I*b/x) - I*gamma(4, -I*b/x))*cos(a) + (gamma(4, I*b/x) + gamma(4, -I*b/x))*sin(a))/b^4

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Fricas [A]  time = 1.53806, size = 112, normalized size = 1.84 \begin{align*} \frac{{\left (b^{3} - 6 \, b x^{2}\right )} \cos \left (\frac{a x + b}{x}\right ) - 3 \,{\left (b^{2} x - 2 \, x^{3}\right )} \sin \left (\frac{a x + b}{x}\right )}{b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x^5,x, algorithm="fricas")

[Out]

((b^3 - 6*b*x^2)*cos((a*x + b)/x) - 3*(b^2*x - 2*x^3)*sin((a*x + b)/x))/(b^4*x^3)

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Sympy [A]  time = 7.66222, size = 61, normalized size = 1. \begin{align*} \begin{cases} \frac{\cos{\left (a + \frac{b}{x} \right )}}{b x^{3}} - \frac{3 \sin{\left (a + \frac{b}{x} \right )}}{b^{2} x^{2}} - \frac{6 \cos{\left (a + \frac{b}{x} \right )}}{b^{3} x} + \frac{6 \sin{\left (a + \frac{b}{x} \right )}}{b^{4}} & \text{for}\: b \neq 0 \\- \frac{\sin{\left (a \right )}}{4 x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x**5,x)

[Out]

Piecewise((cos(a + b/x)/(b*x**3) - 3*sin(a + b/x)/(b**2*x**2) - 6*cos(a + b/x)/(b**3*x) + 6*sin(a + b/x)/b**4,
Ne(b, 0)), (-sin(a)/(4*x**4), 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x)/x^5,x, algorithm="giac")

[Out]

integrate(sin(a + b/x)/x^5, x)