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

Optimal. Leaf size=103 \[ \frac{a \log \left (\frac{e}{x}+f\right )}{f}+\frac{a \log (x)}{f}+\frac{b \sin \left (c-\frac{d f}{e}\right ) \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{f}-\frac{b \sin (c) \text{CosIntegral}\left (\frac{d}{x}\right )}{f}+\frac{b \cos \left (c-\frac{d f}{e}\right ) \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{f}-\frac{b \cos (c) \text{Si}\left (\frac{d}{x}\right )}{f} \]

[Out]

(a*Log[f + e/x])/f + (a*Log[x])/f - (b*CosIntegral[d/x]*Sin[c])/f + (b*CosIntegral[d*(f/e + x^(-1))]*Sin[c - (
d*f)/e])/f + (b*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + x^(-1))])/f - (b*Cos[c]*SinIntegral[d/x])/f

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Rubi [A]  time = 0.278575, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3431, 14, 3303, 3299, 3302, 3317} \[ \frac{a \log \left (\frac{e}{x}+f\right )}{f}+\frac{a \log (x)}{f}+\frac{b \sin \left (c-\frac{d f}{e}\right ) \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{f}-\frac{b \sin (c) \text{CosIntegral}\left (\frac{d}{x}\right )}{f}+\frac{b \cos \left (c-\frac{d f}{e}\right ) \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{f}-\frac{b \cos (c) \text{Si}\left (\frac{d}{x}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d/x])/(e + f*x),x]

[Out]

(a*Log[f + e/x])/f + (a*Log[x])/f - (b*CosIntegral[d/x]*Sin[c])/f + (b*CosIntegral[d*(f/e + x^(-1))]*Sin[c - (
d*f)/e])/f + (b*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + x^(-1))])/f - (b*Cos[c]*SinIntegral[d/x])/f

Rule 3431

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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]

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rubi steps

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

Mathematica [A]  time = 0.201585, size = 83, normalized size = 0.81 \[ \frac{a \log (e+f x)+b \sin \left (c-\frac{d f}{e}\right ) \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )-b \sin (c) \text{CosIntegral}\left (\frac{d}{x}\right )+b \cos \left (c-\frac{d f}{e}\right ) \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )-b \cos (c) \text{Si}\left (\frac{d}{x}\right )}{f} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sin[c + d/x])/(e + f*x),x]

[Out]

(a*Log[e + f*x] - b*CosIntegral[d/x]*Sin[c] + b*CosIntegral[d*(f/e + x^(-1))]*Sin[c - (d*f)/e] + b*Cos[c - (d*
f)/e]*SinIntegral[d*(f/e + x^(-1))] - b*Cos[c]*SinIntegral[d/x])/f

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Maple [A]  time = 0.023, size = 142, normalized size = 1.4 \begin{align*} -{\frac{a}{f}\ln \left ({\frac{d}{x}} \right ) }+{\frac{a}{f}\ln \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) }+{\frac{b}{f}{\it Si} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \cos \left ({\frac{-ce+df}{e}} \right ) }-{\frac{b}{f}{\it Ci} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \sin \left ({\frac{-ce+df}{e}} \right ) }-{\frac{b\cos \left ( c \right ) }{f}{\it Si} \left ({\frac{d}{x}} \right ) }-{\frac{b\sin \left ( c \right ) }{f}{\it Ci} \left ({\frac{d}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b{\left (\int \frac{\sin \left (\frac{c x + d}{x}\right )}{2 \,{\left ({\left (f x + e\right )} \cos \left (\frac{c x + d}{x}\right )^{2} +{\left (f x + e\right )} \sin \left (\frac{c x + d}{x}\right )^{2}\right )}}\,{d x} + \int \frac{\sin \left (\frac{c x + d}{x}\right )}{2 \,{\left (f x + e\right )}}\,{d x}\right )} + \frac{a \log \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*(integrate(1/2*sin((c*x + d)/x)/((f*x + e)*cos((c*x + d)/x)^2 + (f*x + e)*sin((c*x + d)/x)^2), x) + integrat
e(1/2*sin((c*x + d)/x)/(f*x + e), x)) + a*log(f*x + e)/f

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Fricas [A]  time = 1.5374, size = 366, normalized size = 3.55 \begin{align*} -\frac{2 \, b \cos \left (c\right ) \operatorname{Si}\left (\frac{d}{x}\right ) - 2 \, b \cos \left (-\frac{c e - d f}{e}\right ) \operatorname{Si}\left (\frac{d f x + d e}{e x}\right ) - 2 \, a \log \left (f x + e\right ) +{\left (b \operatorname{Ci}\left (\frac{d}{x}\right ) + b \operatorname{Ci}\left (-\frac{d}{x}\right )\right )} \sin \left (c\right ) +{\left (b \operatorname{Ci}\left (\frac{d f x + d e}{e x}\right ) + b \operatorname{Ci}\left (-\frac{d f x + d e}{e x}\right )\right )} \sin \left (-\frac{c e - d f}{e}\right )}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*cos(c)*sin_integral(d/x) - 2*b*cos(-(c*e - d*f)/e)*sin_integral((d*f*x + d*e)/(e*x)) - 2*a*log(f*x +
e) + (b*cos_integral(d/x) + b*cos_integral(-d/x))*sin(c) + (b*cos_integral((d*f*x + d*e)/(e*x)) + b*cos_integ
ral(-(d*f*x + d*e)/(e*x)))*sin(-(c*e - d*f)/e))/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sin(c + d/x))/(e + f*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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