3.105 \(\int \sin (a+\frac {b}{x}) \, dx\)

Optimal. Leaf size=32 \[ -b \cos (a) \text {Ci}\left (\frac {b}{x}\right )+b \sin (a) \text {Si}\left (\frac {b}{x}\right )+x \sin \left (a+\frac {b}{x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {3361, 3297, 3303, 3299, 3302} \[ -b \cos (a) \text {CosIntegral}\left (\frac {b}{x}\right )+b \sin (a) \text {Si}\left (\frac {b}{x}\right )+x \sin \left (a+\frac {b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/x],x]

[Out]

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

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 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]

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 3361

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Sin[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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 32, normalized size = 1.00 \[ -b \cos (a) \text {Ci}\left (\frac {b}{x}\right )+b \sin (a) \text {Si}\left (\frac {b}{x}\right )+x \sin \left (a+\frac {b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/x],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 45, normalized size = 1.41 \[ b \sin \relax (a) \operatorname {Si}\left (\frac {b}{x}\right ) - \frac {1}{2} \, {\left (b \operatorname {Ci}\left (\frac {b}{x}\right ) + b \operatorname {Ci}\left (-\frac {b}{x}\right )\right )} \cos \relax (a) + x \sin \left (\frac {a x + b}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.46, size = 132, normalized size = 4.12 \[ -\frac {a b^{2} \cos \relax (a) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right ) + a b^{2} \sin \relax (a) \operatorname {Si}\left (a - \frac {a x + b}{x}\right ) - \frac {{\left (a x + b\right )} b^{2} \cos \relax (a) \operatorname {Ci}\left (-a + \frac {a x + b}{x}\right )}{x} - \frac {{\left (a x + b\right )} b^{2} \sin \relax (a) \operatorname {Si}\left (a - \frac {a x + b}{x}\right )}{x} + b^{2} \sin \left (\frac {a x + b}{x}\right )}{{\left (a - \frac {a x + b}{x}\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.03, size = 38, normalized size = 1.19 \[ -b \left (-\frac {\sin \left (a +\frac {b}{x}\right ) x}{b}-\Si \left (\frac {b}{x}\right ) \sin \relax (a )+\Ci \left (\frac {b}{x}\right ) \cos \relax (a )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 0.38, size = 58, normalized size = 1.81 \[ -\frac {1}{2} \, {\left ({\left ({\rm Ei}\left (\frac {i \, b}{x}\right ) + {\rm Ei}\left (-\frac {i \, b}{x}\right )\right )} \cos \relax (a) - {\left (-i \, {\rm Ei}\left (\frac {i \, b}{x}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{x}\right )\right )} \sin \relax (a)\right )} b + x \sin \left (\frac {a x + b}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \sin \left (a+\frac {b}{x}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b/x),x)

[Out]

int(sin(a + b/x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\left (a + \frac {b}{x} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b/x), x)

________________________________________________________________________________________