3.290 \(\int (a+b \sin (c+\frac{d}{x})) \, dx\)

Optimal. Leaf size=38 \[ a x-b d \cos (c) \text{CosIntegral}\left (\frac{d}{x}\right )+b d \sin (c) \text{Si}\left (\frac{d}{x}\right )+b x \sin \left (c+\frac{d}{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0783685, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3361, 3297, 3303, 3299, 3302} \[ a x-b d \cos (c) \text{CosIntegral}\left (\frac{d}{x}\right )+b d \sin (c) \text{Si}\left (\frac{d}{x}\right )+b x \sin \left (c+\frac{d}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sin[c + d/x],x]

[Out]

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

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]

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

Mathematica [A]  time = 0.0303449, size = 50, normalized size = 1.32 \[ a x-b d \left (\cos (c) \text{CosIntegral}\left (\frac{d}{x}\right )-\sin (c) \text{Si}\left (\frac{d}{x}\right )\right )+b x \sin (c) \cos \left (\frac{d}{x}\right )+b x \cos (c) \sin \left (\frac{d}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sin[c + d/x],x]

[Out]

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

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Maple [A]  time = 0.01, size = 43, normalized size = 1.1 \begin{align*} ax-bd \left ( -{\frac{x}{d}\sin \left ( c+{\frac{d}{x}} \right ) }-{\it Si} \left ({\frac{d}{x}} \right ) \sin \left ( c \right ) +{\it Ci} \left ({\frac{d}{x}} \right ) \cos \left ( c \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*sin(c+d/x),x)

[Out]

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

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Maxima [C]  time = 1.12897, size = 88, normalized size = 2.32 \begin{align*} -\frac{1}{2} \,{\left ({\left ({\left ({\rm Ei}\left (\frac{i \, d}{x}\right ) +{\rm Ei}\left (-\frac{i \, d}{x}\right )\right )} \cos \left (c\right ) -{\left (-i \,{\rm Ei}\left (\frac{i \, d}{x}\right ) + i \,{\rm Ei}\left (-\frac{i \, d}{x}\right )\right )} \sin \left (c\right )\right )} d - 2 \, x \sin \left (\frac{c x + d}{x}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.30836, size = 163, normalized size = 4.29 \begin{align*} b d \sin \left (c\right ) \operatorname{Si}\left (\frac{d}{x}\right ) + b x \sin \left (\frac{c x + d}{x}\right ) + a x - \frac{1}{2} \,{\left (b d \operatorname{Ci}\left (\frac{d}{x}\right ) + b d \operatorname{Ci}\left (-\frac{d}{x}\right )\right )} \cos \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sin(c+d/x),x)

[Out]

Integral(a + b*sin(c + d/x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*sin(c + d/x) + a, x)