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:
Ci(b*c/d+b*x)*sin(a-b*c/d)/d+cos(a-b*c/d)*Si(b*c/d+b*x)/d
Time = 0.13 (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
Time = 0.38 (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
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]
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]
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]
Time = 0.72 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.53
method | result | size |
derivativedivides | \(-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}\) | \(78\) |
default | \(-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}\) | \(78\) |
risch | \(\frac {i {\mathrm e}^{\frac {i \left (a d -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (-i b x -i a -\frac {-i a d +i b c}{d}\right )}{2 d}-\frac {i {\mathrm e}^{-\frac {i \left (a d -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (i b x +i a -\frac {i \left (a d -b c \right )}{d}\right )}{2 d}\) | \(98\) |
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
Time = 0.08 (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
\[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )}}{c + d x}\, dx \] Input:
integrate(sin(b*x+a)/(d*x+c),x)
Output:
Integral(sin(a + b*x)/(c + d*x), x)
Result contains complex when optimal does not.
Time = 0.08 (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)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.42 (sec) , antiderivative size = 597, normalized size of antiderivative = 11.71 \[ \int \frac {\sin (a+b x)}{c+d x} \, dx =\text {Too large to display} \] 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)
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)
\[ \int \frac {\sin (a+b x)}{c+d x} \, dx=\int \frac {\sin \left (b x +a \right )}{d x +c}d x \] Input:
int(sin(b*x+a)/(d*x+c),x)
Output:
int(sin(a + b*x)/(c + d*x),x)