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

Optimal. Leaf size=73 \[ -\frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a}+\frac{\sin (c) \text{CosIntegral}(d x)}{a}+\frac{\cos (c) \text{Si}(d x)}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.261002, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6742, 3303, 3299, 3302} \[ -\frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a}+\frac{\sin (c) \text{CosIntegral}(d x)}{a}+\frac{\cos (c) \text{Si}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x)),x]

[Out]

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[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]

Rubi steps

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

Mathematica [A]  time = 0.166073, size = 63, normalized size = 0.86 \[ \frac{-\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )-\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )+\sin (c) \text{CosIntegral}(d x)+\cos (c) \text{Si}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 99, normalized size = 1.4 \begin{align*} -{\frac{b}{a} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) }+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*b*(Si(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b)+1/a*(Si(d*x)*cos(c)
+Ci(d*x)*sin(c))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.74711, size = 306, normalized size = 4.19 \begin{align*} \frac{{\left (\operatorname{Ci}\left (d x\right ) + \operatorname{Ci}\left (-d x\right )\right )} \sin \left (c\right ) +{\left (\operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) + \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right ) + 2 \, \cos \left (c\right ) \operatorname{Si}\left (d x\right ) - 2 \, \cos \left (-\frac{b c - a d}{b}\right ) \operatorname{Si}\left (\frac{b d x + a d}{b}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [C]  time = 1.20304, size = 1131, normalized size = 15.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + imag_part(cos_integral(d*x))*tan(1/
2*c)^2*tan(1/2*a*d/b)^2 - imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - imag_part(cos_
integral(-d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*sin_integral(d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*sin_inte
gral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(
1/2*a*d/b) + 2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*real_part(cos_integral(d*
x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*real_part(cos_integral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*real_part(cos_integral(-d*x))*tan(1/2*c)*tan(
1/2*a*d/b)^2 - imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 + imag_part(cos_integral(d*x))*tan(1/2*c)^2 +
imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*sin_integ
ral(d*x)*tan(1/2*c)^2 - 2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 4*imag_part(cos_integral(d*x + a*d/b))*
tan(1/2*c)*tan(1/2*a*d/b) - 4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*sin_integral
((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - imag_par
t(cos_integral(d*x))*tan(1/2*a*d/b)^2 + imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + imag_part(cos
_integral(-d*x))*tan(1/2*a*d/b)^2 - 2*sin_integral(d*x)*tan(1/2*a*d/b)^2 - 2*sin_integral((b*d*x + a*d)/b)*tan
(1/2*a*d/b)^2 + 2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) - 2*real_part(cos_integral(d*x))*tan(1/2*c)
+ 2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*real_part(cos_integral(-d*x))*tan(1/2*c) - 2*real_par
t(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + imag_pa
rt(cos_integral(d*x + a*d/b)) - imag_part(cos_integral(d*x)) - imag_part(cos_integral(-d*x - a*d/b)) + imag_pa
rt(cos_integral(-d*x)) - 2*sin_integral(d*x) + 2*sin_integral((b*d*x + a*d)/b))/(a*tan(1/2*c)^2*tan(1/2*a*d/b)
^2 + a*tan(1/2*c)^2 + a*tan(1/2*a*d/b)^2 + a)