3.35 \(\int \cos (a+\frac{b}{x}) \, dx\)

Optimal. Leaf size=31 \[ b \sin (a) \text{CosIntegral}\left (\frac{b}{x}\right )+b \cos (a) \text{Si}\left (\frac{b}{x}\right )+x \cos \left (a+\frac{b}{x}\right ) \]

[Out]

x*Cos[a + b/x] + b*CosIntegral[b/x]*Sin[a] + b*Cos[a]*SinIntegral[b/x]

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Rubi [A]  time = 0.0723927, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3362, 3297, 3303, 3299, 3302} \[ b \sin (a) \text{CosIntegral}\left (\frac{b}{x}\right )+b \cos (a) \text{Si}\left (\frac{b}{x}\right )+x \cos \left (a+\frac{b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b/x],x]

[Out]

x*Cos[a + b/x] + b*CosIntegral[b/x]*Sin[a] + b*Cos[a]*SinIntegral[b/x]

Rule 3362

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Cos[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && In
tegerQ[1/n]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.030921, size = 31, normalized size = 1. \[ b \sin (a) \text{CosIntegral}\left (\frac{b}{x}\right )+b \cos (a) \text{Si}\left (\frac{b}{x}\right )+x \cos \left (a+\frac{b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b/x],x]

[Out]

x*Cos[a + b/x] + b*CosIntegral[b/x]*Sin[a] + b*Cos[a]*SinIntegral[b/x]

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Maple [A]  time = 0.037, size = 39, normalized size = 1.3 \begin{align*} -b \left ( -{\frac{x}{b}\cos \left ( a+{\frac{b}{x}} \right ) }-{\it Si} \left ({\frac{b}{x}} \right ) \cos \left ( a \right ) -{\it Ci} \left ({\frac{b}{x}} \right ) \sin \left ( a \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(-cos(a+b/x)*x/b-Si(b/x)*cos(a)-Ci(b/x)*sin(a))

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Maxima [C]  time = 1.28664, size = 77, normalized size = 2.48 \begin{align*} \frac{1}{2} \,{\left ({\left (-i \,{\rm Ei}\left (\frac{i \, b}{x}\right ) + i \,{\rm Ei}\left (-\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) +{\left ({\rm Ei}\left (\frac{i \, b}{x}\right ) +{\rm Ei}\left (-\frac{i \, b}{x}\right )\right )} \sin \left (a\right )\right )} b + x \cos \left (\frac{a x + b}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59511, size = 144, normalized size = 4.65 \begin{align*} b \cos \left (a\right ) \operatorname{Si}\left (\frac{b}{x}\right ) + x \cos \left (\frac{a x + b}{x}\right ) + \frac{1}{2} \,{\left (b \operatorname{Ci}\left (\frac{b}{x}\right ) + b \operatorname{Ci}\left (-\frac{b}{x}\right )\right )} \sin \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (a + \frac{b}{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b/x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(a + b/x), x)