3.1.5 \(\int \frac {\sin (a+b x)}{c+d x} \, dx\) [5]

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

[Out]

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

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Rubi [A]
time = 0.07, 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 = {3384, 3380, 3383} \begin {gather*} \frac {\sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]/(c + d*x),x]

[Out]

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

Rule 3380

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 3383

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 3384

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

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Mathematica [A]
time = 0.06, size = 49, normalized size = 0.96 \begin {gather*} \frac {\text {Ci}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )+\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]/(c + d*x),x]

[Out]

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

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Maple [A]
time = 0.05, size = 78, normalized size = 1.53

method	result	size
derivativedivides	\(-\frac {\sinIntegral \left (-b x -a -\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}-\frac {\cosineIntegral \left (b x +a +\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}\)	\(78\)
default	\(-\frac {\sinIntegral \left (-b x -a -\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}-\frac {\cosineIntegral \left (b x +a +\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}\)	\(78\)
risch	\(\frac {i {\mathrm e}^{\frac {i \left (d a -c b \right )}{d}} \expIntegral \left (1, -i b x -i a -\frac {-i a d +i b c}{d}\right )}{2 d}-\frac {i {\mathrm e}^{-\frac {i \left (d a -c b \right )}{d}} \expIntegral \left (1, i b x +i a -\frac {i \left (d a -c b \right )}{d}\right )}{2 d}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.38, size = 141, normalized size = 2.76 \begin {gather*} -\frac {b {\left (i \, E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b {\left (E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 78, normalized size = 1.53 \begin {gather*} \frac {{\left (\operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) + \operatorname {Ci}\left (-\frac {b d x + b c}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + 2 \, \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*x)/(c + d*x), x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 4.29, size = 597, normalized size = 11.71 \begin {gather*} \frac {\Im \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {b c}{2 \, d}\right )^{2} - \Im \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {b c}{2 \, d}\right )^{2} + 2 \, \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {b c}{2 \, d}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {b c}{2 \, d}\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {b c}{2 \, d}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {b c}{2 \, d}\right )^{2} - 2 \, \Re \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {b c}{2 \, d}\right )^{2} - \Im \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \Im \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \Im \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {b c}{2 \, d}\right ) - 4 \, \Im \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {b c}{2 \, d}\right ) + 8 \, \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {b c}{2 \, d}\right ) - \Im \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {b c}{2 \, d}\right )^{2} + \Im \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {b c}{2 \, d}\right )^{2} - 2 \, \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) \tan \left (\frac {b c}{2 \, d}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) \tan \left (\frac {b c}{2 \, d}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) \tan \left (\frac {b c}{2 \, d}\right ) + \Im \left ( \operatorname {Ci}\left (b x + \frac {b c}{d}\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-b x - \frac {b c}{d}\right ) \right ) + 2 \, \operatorname {Si}\left (\frac {b d x + b c}{d}\right )}{2 \, {\left (d \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {b c}{2 \, d}\right )^{2} + d \tan \left (\frac {1}{2} \, a\right )^{2} + d \tan \left (\frac {b c}{2 \, d}\right )^{2} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(-b*x - b*c/d)
)*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*real_part(
cos_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d) + 2*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*
tan(1/2*b*c/d) - 2*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a)*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral
(-b*x - b*c/d))*tan(1/2*a)*tan(1/2*b*c/d)^2 - imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2 + imag_part(co
s_integral(-b*x - b*c/d))*tan(1/2*a)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)^2 + 4*imag_part(cos_integr
al(b*x + b*c/d))*tan(1/2*a)*tan(1/2*b*c/d) - 4*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)*tan(1/2*b*c/d)
+ 8*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)*tan(1/2*b*c/d) - imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*
c/d)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*c/d)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*c/
d)^2 + 2*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a) + 2*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)
- 2*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*c/d) - 2*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*c/
d) + imag_part(cos_integral(b*x + b*c/d)) - imag_part(cos_integral(-b*x - b*c/d)) + 2*sin_integral((b*d*x + b*
c)/d))/(d*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + d*tan(1/2*a)^2 + d*tan(1/2*b*c/d)^2 + d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)/(c + d*x),x)

[Out]

int(sin(a + b*x)/(c + d*x), x)

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