Integrand size = 20, antiderivative size = 65 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )+\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
(CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d] + Cos[2*a - (2*b*c)/d ]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d)
Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4906, 27, 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.
| \(\displaystyle \int \frac {\sin (a+b x) \cos (a+b x)}{c+d x} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \frac {\sin (2 a+2 b x)}{2 (c+d x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x}dx\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {1}{2} \left (\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx+\cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x+\frac {\pi }{2}\right )}{c+d x}dx+\cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {1}{2} \left (\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )\) |
((CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d + (Cos[2*a - (2*b *c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d)/2
3.1.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Time = 0.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(-\frac {\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}-\frac {\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}\) | \(84\) |
| default | \(-\frac {\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}-\frac {\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}\) | \(84\) |
| risch | \(-\frac {i {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{4 d}+\frac {i {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{4 d}\) | \(98\) |
-1/2*Si(-2*x*b-2*a-2*(-a*d+b*c)/d)*cos(2*(-a*d+b*c)/d)/d-1/2*Ci(2*x*b+2*a+ 2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d
Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{2 \, d} \]
1/2*(cos_integral(2*(b*d*x + b*c)/d)*sin(-2*(b*c - a*d)/d) + cos(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d))/d
\[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{c + d x}\, dx \]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.20 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=-\frac {b {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, b d} \]
-1/4*(b*(-I*exp_integral_e(1, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*ex p_integral_e(1, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d) /d) + b*(exp_integral_e(1, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_int egral_e(1, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a*d)/d))/ (b*d)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.31 (sec) , antiderivative size = 569, normalized size of antiderivative = 8.75 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} - \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right )^{2} - 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right )^{2} - \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} + \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} - 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right )^{2} + 4 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) - 4 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) + 8 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) - \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} + \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} - 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) + \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) + 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{4 \, {\left (d \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + d \tan \left (a\right )^{2} + d \tan \left (\frac {b c}{d}\right )^{2} + d\right )}} \]
1/4*(imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - imag _part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*sin_integr al(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(2*b *x + 2*b*c/d))*tan(a)^2*tan(b*c/d) + 2*real_part(cos_integral(-2*b*x - 2*b *c/d))*tan(a)^2*tan(b*c/d) - 2*real_part(cos_integral(2*b*x + 2*b*c/d))*ta n(a)*tan(b*c/d)^2 - 2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan (b*c/d)^2 - imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 + imag_part( cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 - 2*sin_integral(2*(b*d*x + b*c)/ d)*tan(a)^2 + 4*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) - 4*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) + 8*sin_i ntegral(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d) - imag_part(cos_integral(2*b* x + 2*b*c/d))*tan(b*c/d)^2 + imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan (b*c/d)^2 - 2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 + 2*real_part(c os_integral(2*b*x + 2*b*c/d))*tan(a) + 2*real_part(cos_integral(-2*b*x - 2 *b*c/d))*tan(a) - 2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + imag_part(cos_int egral(2*b*x + 2*b*c/d)) - imag_part(cos_integral(-2*b*x - 2*b*c/d)) + 2*si n_integral(2*(b*d*x + b*c)/d))/(d*tan(a)^2*tan(b*c/d)^2 + d*tan(a)^2 + d*t an(b*c/d)^2 + d)
Timed out. \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )}{c+d\,x} \,d x \]