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

3.1.5.1 Optimal result

Integrand size = 14, antiderivative size = 51 \[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \]

output

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

3.1.5.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )+\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \]

input

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

output

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

3.1.5.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 3784, 3042, 3780, 3783}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

\(\displaystyle \sin \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x}dx+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{c+d x}dx+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 3780

\(\displaystyle \sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}\)

input

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

output

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

3.1.5.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
 

3.1.5.4 Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(-\frac {\operatorname {Si}\left (-b x -a -\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}\)	\(78\)
default	\(-\frac {\operatorname {Si}\left (-b x -a -\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}\)	\(78\)
risch	\(\frac {i {\mathrm e}^{\frac {i \left (d a -c b \right )}{d}} \operatorname {Ei}_{1}\left (-i b x -i a -\frac {-i a d +i c b}{d}\right )}{2 d}-\frac {i {\mathrm e}^{-\frac {i \left (d a -c b \right )}{d}} \operatorname {Ei}_{1}\left (i b x +i a -\frac {i \left (d a -c b \right )}{d}\right )}{2 d}\)	\(98\)

input

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

output

-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
 

3.1.5.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20 \[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right )}{d} \]

input

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

output

(cos_integral((b*d*x + b*c)/d)*sin(-(b*c - a*d)/d) + cos(-(b*c - a*d)/d)*s 
in_integral((b*d*x + b*c)/d))/d
 

3.1.5.6 Sympy [F]

\[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )}}{c + d x}\, dx \]

input

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

output

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

3.1.5.7 Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.76 \[ \int \frac {\sin (a+b x)}{c+d x} \, dx=-\frac {b {\left (i \, E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b {\left (E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, b d} \]

input

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

output

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

3.1.5.8 Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.30 (sec) , antiderivative size = 597, normalized size of antiderivative = 11.71 \[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\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 )}} \]

input

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

output

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*si 
n_integral((b*d*x + b*c)/d)*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*real_part(co 
s_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d) + 2*real_part(cos_int 
egral(-b*x - b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d) - 2*real_part(cos_integra 
l(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(cos_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_integral(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_p 
art(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*ta 
n(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)
 

3.1.5.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\int \frac {\sin \left (a+b\,x\right )}{c+d\,x} \,d x \]

input

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

output

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