\(\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx\) [6]

Optimal result

Integrand size = 20, antiderivative size = 65 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]

[Out]

1/2*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d+1/2*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4491, 12, 3384, 3380, 3383} \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]

[In]

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

[Out]

(CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(2*d) + (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*
b*x])/(2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx \\ & = \frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx \\ & = \frac {1}{2} \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+\frac {1}{2} \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 \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )+\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

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	\(-\frac {i {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{4 d}+\frac {i {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{4 d}\)	\(98\)

[In]

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

[Out]

-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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{2 \, d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.20 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=-\frac {b {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, b d} \]

[In]

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

[Out]

-1/4*(b*(-I*exp_integral_e(1, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_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_i
ntegral_e(1, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a*d)/d))/(b*d)

Giac [C] (verification not implemented)

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 )}} \]

[In]

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

[Out]

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_integral(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*rea
l_part(cos_integral(2*b*x + 2*b*c/d))*tan(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_integral(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(cos_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_integral(2*b*x + 2*b*c/d)) -
imag_part(cos_integral(-2*b*x - 2*b*c/d)) + 2*sin_integral(2*(b*d*x + b*c)/d))/(d*tan(a)^2*tan(b*c/d)^2 + d*ta
n(a)^2 + d*tan(b*c/d)^2 + d)

Mupad [F(-1)]

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

[In]

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

[Out]

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