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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 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]

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*}

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Mathematica [A]  time = 0.07, size = 49, normalized size = 0.96 \[ \frac {\sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )+\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b} \]

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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fricas [A]  time = 0.63, size = 78, normalized size = 1.53 \[ -\frac {{\left (\operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right ) - 2 \, \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right )}{2 \, b} \]

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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giac [C]  time = 1.53, size = 597, normalized size = 11.71 \[ \frac {\Im \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} - \Im \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + 2 \, \Re \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right ) + 2 \, \Re \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right ) - 2 \, \Re \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - 2 \, \Re \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - \Im \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \Im \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, \Im \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right ) - 4 \, \Im \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right ) + 8 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right ) - \Im \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} + \Im \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} + 2 \, \Re \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, \Re \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, \Re \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right ) - 2 \, \Re \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right ) + \Im \left (\operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) - \Im \left (\operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) + 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right )}{2 \, {\left (b \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + b \tan \left (\frac {1}{2} \, c\right )^{2} + b \tan \left (\frac {a d}{2 \, b}\right )^{2} + b\right )}} \]

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)

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maple [A]  time = 0.02, size = 73, normalized size = 1.43 \[ \frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b} \]

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 = 0.72, size = 141, normalized size = 2.76 \[ \frac {d {\left (-i \, E_{1}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{1}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + d {\left (E_{1}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{1}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, b d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (c+d\,x\right )}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(a + b*x),x)

[Out]

int(sin(c + d*x)/(a + b*x), x)

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

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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