\(\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx\) [7]
Optimal result
Integrand size = 20, antiderivative size = 85 \[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=\frac {b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {\sin (2 a+2 b x)}{2 d (c+d x)}-\frac {b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}
\]
[Out]
b*Ci(2*b*c/d+2*b*x)*cos(2*a-2*b*c/d)/d^2-b*Si(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^2-1/2*sin(2*b*x+2*a)/d/(d*x+c)
Rubi [A] (verified)
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4491, 12, 3378, 3384, 3380,
3383} \[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=\frac {b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {\sin (2 a+2 b x)}{2 d (c+d x)}
\]
[In]
Int[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x)^2,x]
[Out]
(b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^2 - Sin[2*a + 2*b*x]/(2*d*(c + d*x)) - (b*Sin[2*a -
(2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 3378
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]
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)^2} \, dx \\ & = \frac {1}{2} \int \frac {\sin (2 a+2 b x)}{(c+d x)^2} \, dx \\ & = -\frac {\sin (2 a+2 b x)}{2 d (c+d x)}+\frac {b \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx}{d} \\ & = -\frac {\sin (2 a+2 b x)}{2 d (c+d x)}+\frac {\left (b \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}-\frac {\left (b \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d} \\ & = \frac {b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {\sin (2 a+2 b x)}{2 d (c+d x)}-\frac {b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94
\[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=\frac {2 b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right )-\frac {d \sin (2 (a+b x))}{c+d x}-2 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{2 d^2}
\]
[In]
Integrate[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x)^2,x]
[Out]
(2*b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - (d*Sin[2*(a + b*x)])/(c + d*x) - 2*b*Sin[2*a - (2*b
*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/(2*d^2)
Maple [A] (verified)
Time = 0.43 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.46
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {b \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{4}\) |
\(124\) |
| default |
\(\frac {b \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{4}\) |
\(124\) |
| risch |
\(-\frac {b \,{\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{2 d^{2}}-\frac {\left (-2 d x b -2 c b \right ) \sin \left (2 x b +2 a \right )}{4 d \left (-d x b -c b \right ) \left (d x +c \right )}\) |
\(141\) |
| | |
|
|
|
|
[In]
int(cos(b*x+a)*sin(b*x+a)/(d*x+c)^2,x,method=_RETURNVERBOSE)
[Out]
1/4*b*(-2*sin(2*b*x+2*a)/(-a*d+c*b+d*(b*x+a))/d+2*(-2*Si(-2*x*b-2*a-2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d+2*Ci
(2*x*b+2*a+2*(-a*d+b*c)/d)*cos(2*(-a*d+b*c)/d)/d)/d)
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.24
\[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=\frac {{\left (b d x + b c\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - {\left (b d x + b c\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{d^{3} x + c d^{2}}
\]
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)^2,x, algorithm="fricas")
[Out]
((b*d*x + b*c)*cos(-2*(b*c - a*d)/d)*cos_integral(2*(b*d*x + b*c)/d) - d*cos(b*x + a)*sin(b*x + a) - (b*d*x +
b*c)*sin(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d))/(d^3*x + c*d^2)
Sympy [F]
\[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx
\]
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)**2,x)
[Out]
Integral(sin(a + b*x)*cos(a + b*x)/(c + d*x)**2, x)
Maxima [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.95
\[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=-\frac {b^{2} {\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b}
\]
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)^2,x, algorithm="maxima")
[Out]
-1/4*(b^2*(-I*exp_integral_e(2, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(2, -2*(-I*b*c - I*(b*
x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d)/d) + b^2*(exp_integral_e(2, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + e
xp_integral_e(2, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a*d)/d))/((b*c*d + (b*x + a)*d^2 - a*d^
2)*b)
Giac [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.51 (sec) , antiderivative size = 2870, normalized size of antiderivative = 33.76
\[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(b*x+a)*sin(b*x+a)/(d*x+c)^2,x, algorithm="giac")
[Out]
1/2*(b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + b*d*x*real_part(cos_int
egral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*t
an(b*x)^2*tan(a)^2*tan(b*c/d) + 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/
d) - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 2*b*d*x*imag_part(cos_integral(2
*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2
*tan(a)*tan(b*c/d)^2 + 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + b*c*real_part(
cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + b*c*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 - b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 - b*d*
x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 + 4*b*d*x*real_part(cos_integral(2*b*x + 2*b*c
/d))*tan(b*x)^2*tan(a)*tan(b*c/d) + 4*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*
c/d) - 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 2*b*c*imag_part(cos_int
egral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 4*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan
(a)^2*tan(b*c/d) - b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - b*d*x*real_part(co
s_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x
)^2*tan(a)*tan(b*c/d)^2 - 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 4*b
*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + b*d*x*real_part(cos_integral(2*b*x + 2*b*c
/d))*tan(a)^2*tan(b*c/d)^2 + b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - 2*b*d*x*i
mag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a) + 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(b*x)^2*tan(a) - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a) - b*c*real_part(cos_integral(2*b
*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 - b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 + 2*b*d
*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*
b*c/d))*tan(b*x)^2*tan(b*c/d) + 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d) + 4*b*c*real_par
t(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b*c/d) + 4*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(b*x)^2*tan(a)*tan(b*c/d) - 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) + 2*b*d*
x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(
a)^2*tan(b*c/d) - b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - b*c*real_part(cos_int
egral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan
(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 + 4*b*d*x*sin_integral(2*(b*
d*x + b*c)/d)*tan(a)*tan(b*c/d)^2 + b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + b*c*r
eal_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 + b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d)
)*tan(b*x)^2 + b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 - 2*b*c*imag_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(b*x)^2*tan(a) + 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a) - 4*b*c*sin
_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a) - b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 - b*d
*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 + 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x
)^2*tan(b*c/d) - 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 4*b*c*sin_integral(2*
(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d) + 4*b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) +
4*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - 2*b*c*imag_part(cos_integral(2*b*x + 2*b
*c/d))*tan(a)^2*tan(b*c/d) + 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) - 4*b*c*sin_i
ntegral(2*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d) - b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 -
b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 + 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))
*tan(a)*tan(b*c/d)^2 - 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 + 4*b*c*sin_integra
l(2*(b*d*x + b*c)/d)*tan(a)*tan(b*c/d)^2 + 2*d*tan(b*x)^2*tan(a)*tan(b*c/d)^2 + 2*d*tan(b*x)*tan(a)^2*tan(b*c/
d)^2 + b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 + b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))
*tan(b*x)^2 - 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) + 2*b*d*x*imag_part(cos_integral(-2*b*x
- 2*b*c/d))*tan(a) - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(a) - b*c*real_part(cos_integral(2*b*x + 2*b*c
/d))*tan(a)^2 - b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2 + 2*b*d*x*imag_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(b*c/d) - 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 4*b*d*x*sin_integral(2
*(b*d*x + b*c)/d)*tan(b*c/d) + 4*b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d) + 4*b*c*real_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d
)^2 - b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 + b*d*x*real_part(cos_integral(2*b*x + 2*b*c/
d)) + b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d)) - 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)
+ 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a) - 4*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(a) + 2*d*
tan(b*x)^2*tan(a) + 2*d*tan(b*x)*tan(a)^2 + 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) - 2*b*c*
imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) + 4*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d) - 2*d*
tan(b*x)*tan(b*c/d)^2 - 2*d*tan(a)*tan(b*c/d)^2 + b*c*real_part(cos_integral(2*b*x + 2*b*c/d)) + b*c*real_part
(cos_integral(-2*b*x - 2*b*c/d)) - 2*d*tan(b*x) - 2*d*tan(a))/(d^3*x*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + c*d^2*
tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + d^3*x*tan(b*x)^2*tan(a)^2 + d^3*x*tan(b*x)^2*tan(b*c/d)^2 + d^3*x*tan(a)^2*
tan(b*c/d)^2 + c*d^2*tan(b*x)^2*tan(a)^2 + c*d^2*tan(b*x)^2*tan(b*c/d)^2 + c*d^2*tan(a)^2*tan(b*c/d)^2 + d^3*x
*tan(b*x)^2 + d^3*x*tan(a)^2 + d^3*x*tan(b*c/d)^2 + c*d^2*tan(b*x)^2 + c*d^2*tan(a)^2 + c*d^2*tan(b*c/d)^2 + d
^3*x + c*d^2)
Mupad [F(-1)]
Timed out. \[
\int \frac {\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x
\]
[In]
int((cos(a + b*x)*sin(a + b*x))/(c + d*x)^2,x)
[Out]
int((cos(a + b*x)*sin(a + b*x))/(c + d*x)^2, x)