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

Optimal. Leaf size=51 \[ \frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b}+\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0782072, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3303, 3299, 3302} \[ \frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b}+\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*x),x]

[Out]

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

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

Mathematica [A]  time = 0.07534, size = 49, normalized size = 0.96 \[ \frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )+\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.007, size = 73, normalized size = 1.4 \begin{align*}{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.18376, size = 190, normalized size = 3.73 \begin{align*} \frac{d{\left (-i \, E_{1}\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{1}\left (-\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) + d{\left (E_{1}\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{1}\left (-\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{2 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.657, size = 201, normalized size = 3.94 \begin{align*} -\frac{{\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 (-\frac{b c - a d}{b}\right ) \operatorname{Si}\left (\frac{b d x + a d}{b}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(c + d*x)/(a + b*x), x)

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Giac [C]  time = 1.16813, size = 806, normalized size = 15.8 \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)/(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 - a*d/b)
)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*sin_integral((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 - a*d/b))*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(co
s_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 4*imag_part(cos_integr
al(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_part(cos_integral(-d*x - a*d/b))*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 - a*d/b))*tan(1/2*c)
- 2*real_part(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_part(cos_integral(d*x + a*d/b)) - imag_part(cos_integral(-d*x - a*d/b)) + 2*sin_integral((b*d*x + a*
d)/b))/(b*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b*tan(1/2*c)^2 + b*tan(1/2*a*d/b)^2 + b)