\(\int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx\) [85]

Optimal result

Integrand size = 24, antiderivative size = 104 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=-\frac {1}{8 d (c+d x)}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}+\frac {b \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{2 d^2}+\frac {b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2} \]

[Out]

-1/8/d/(d*x+c)+1/8*cos(4*b*x+4*a)/d/(d*x+c)+1/2*b*cos(4*a-4*b*c/d)*Si(4*b*c/d+4*b*x)/d^2+1/2*b*Ci(4*b*c/d+4*b*
x)*sin(4*a-4*b*c/d)/d^2

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4491, 3378, 3384, 3380, 3383} \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=\frac {b \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2}+\frac {b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}-\frac {1}{8 d (c+d x)} \]

[In]

Int[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^2,x]

[Out]

-1/8*1/(d*(c + d*x)) + Cos[4*a + 4*b*x]/(8*d*(c + d*x)) + (b*CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/
d])/(2*d^2) + (b*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(2*d^2)

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 \left (\frac {1}{8 (c+d x)^2}-\frac {\cos (4 a+4 b x)}{8 (c+d x)^2}\right ) \, dx \\ & = -\frac {1}{8 d (c+d x)}-\frac {1}{8} \int \frac {\cos (4 a+4 b x)}{(c+d x)^2} \, dx \\ & = -\frac {1}{8 d (c+d x)}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}+\frac {b \int \frac {\sin (4 a+4 b x)}{c+d x} \, dx}{2 d} \\ & = -\frac {1}{8 d (c+d x)}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}+\frac {\left (b \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx}{2 d}+\frac {\left (b \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx}{2 d} \\ & = -\frac {1}{8 d (c+d x)}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}+\frac {b \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{2 d^2}+\frac {b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=\frac {\frac {d (-1+\cos (4 (a+b x)))}{c+d x}+4 b \operatorname {CosIntegral}\left (\frac {4 b (c+d x)}{d}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )+4 b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )}{8 d^2} \]

[In]

Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^2,x]

[Out]

((d*(-1 + Cos[4*(a + b*x)]))/(c + d*x) + 4*b*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + 4*b*Cos[4*a
- (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(8*d^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.49

method	result	size
risch	\(-\frac {1}{8 d \left (d x +c \right )}-\frac {i b \,{\mathrm e}^{-\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (4 i b x +4 i a -\frac {4 i \left (a d -c b \right )}{d}\right )}{4 d^{2}}+\frac {i b \,{\mathrm e}^{\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-4 i b x -4 i a -\frac {4 \left (-i a d +i c b \right )}{d}\right )}{4 d^{2}}+\frac {\left (-2 d x b -2 c b \right ) \cos \left (4 x b +4 a \right )}{16 d \left (-d x b -c b \right ) \left (d x +c \right )}\)	\(155\)
derivativedivides	\(\frac {-\frac {b^{2} \left (-\frac {4 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {4 \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{d}\right )}{32}-\frac {b^{2}}{8 \left (-a d +c b +d \left (x b +a \right )\right ) d}}{b}\)	\(156\)
default	\(\frac {-\frac {b^{2} \left (-\frac {4 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {4 \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{d}\right )}{32}-\frac {b^{2}}{8 \left (-a d +c b +d \left (x b +a \right )\right ) d}}{b}\)	\(156\)

[In]

int(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

-1/8/d/(d*x+c)-1/4*I*b/d^2*exp(-4*I*(a*d-b*c)/d)*Ei(1,4*I*b*x+4*I*a-4*I*(a*d-b*c)/d)+1/4*I*b/d^2*exp(4*I*(a*d-
b*c)/d)*Ei(1,-4*I*x*b-4*I*a-4*(-I*a*d+I*c*b)/d)+1/16/d*(-2*b*d*x-2*b*c)/(-b*d*x-b*c)/(d*x+c)*cos(4*b*x+4*a)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=\frac {2 \, d \cos \left (b x + a\right )^{4} - 2 \, d \cos \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + {\left (b d x + b c\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

1/2*(2*d*cos(b*x + a)^4 - 2*d*cos(b*x + a)^2 + (b*d*x + b*c)*cos_integral(4*(b*d*x + b*c)/d)*sin(-4*(b*c - a*d
)/d) + (b*d*x + b*c)*cos(-4*(b*c - a*d)/d)*sin_integral(4*(b*d*x + b*c)/d))/(d^3*x + c*d^2)

Sympy [F]

\[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(cos(b*x+a)**2*sin(b*x+a)**2/(d*x+c)**2,x)

[Out]

Integral(sin(a + b*x)**2*cos(a + b*x)**2/(c + d*x)**2, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.64 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=\frac {b^{2} {\left (E_{2}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{2}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + b^{2} {\left (i \, E_{2}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{2}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, b^{2}}{16 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

1/16*(b^2*(exp_integral_e(2, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(2, -4*(-I*b*c - I*(b*x + a
)*d + I*a*d)/d))*cos(-4*(b*c - a*d)/d) + b^2*(I*exp_integral_e(2, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I*ex
p_integral_e(2, -4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-4*(b*c - a*d)/d) - 2*b^2)/((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 = 0.76 (sec) , antiderivative size = 3218, normalized size of antiderivative = 30.94 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

1/4*(b*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - b*d*x*imag_part(c
os_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 2*b*d*x*sin_integral(4*(b*d*x + b*c)/d
)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2
*a)^2*tan(2*b*c/d) + 2*b*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) -
2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 2*b*d*x*real_part(cos_
integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + b*c*imag_part(cos_integral(4*b*x + 4*b*c/d)
)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)
^2*tan(2*b*c/d)^2 + 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - b*d*x*imag_
part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*
tan(2*b*x)^2*tan(2*a)^2 - 2*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2 + 4*b*d*x*imag_part(
cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 4*b*d*x*imag_part(cos_integral(-4*b*x - 4*
b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 8*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*tan
(2*b*c/d) + 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 2*b*c*real_p
art(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - b*d*x*imag_part(cos_integral(4*b*x
+ 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b
*c/d)^2 - 2*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(4
*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*
b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - b*
d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 2*b*d*x*sin_integral(4*(b*d*x + b*c)
/d)*tan(2*a)^2*tan(2*b*c/d)^2 + 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a) + 2*b*d
*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - b*c*imag_part(cos_integral(4*b*x + 4*b*c/
d))*tan(2*b*x)^2*tan(2*a)^2 + b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 - 2*b*c*si
n_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2 - 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2
*b*x)^2*tan(2*b*c/d) - 2*b*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 4*b*c*ima
g_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 4*b*c*imag_part(cos_integral(-4*b*x
- 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 8*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*
tan(2*b*c/d) + 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 2*b*d*x*real_part(co
s_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - b*c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x
)^2*tan(2*b*c/d)^2 + b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 - 2*b*c*sin_int
egral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d)^2 - 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*
a)*tan(2*b*c/d)^2 - 2*b*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + b*c*imag_part(
cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2
*a)^2*tan(2*b*c/d)^2 + 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_i
ntegral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 - b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 + 2*b*d*
x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2 + 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*t
an(2*a) + 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - b*d*x*imag_part(cos_integral
(4*b*x + 4*b*c/d))*tan(2*a)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 2*b*d*x*sin_integ
ral(4*(b*d*x + b*c)/d)*tan(2*a)^2 - 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) -
2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 4*b*d*x*imag_part(cos_integral(4*
b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) - 4*b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d
) + 8*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d) + 2*b*c*real_part(cos_integral(4*b*x + 4*b*c
/d))*tan(2*a)^2*tan(2*b*c/d) + 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - b*d*x
*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*b*c/d)^2 - 2*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(4*b*x + 4*
b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + b*
c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 - b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*
b*x)^2 + 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2 + 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))
*tan(2*a) + 2*b*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - b*c*imag_part(cos_integral(4*b*x + 4*
b*c/d))*tan(2*a)^2 + b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 2*b*c*sin_integral(4*(b*d*x +
b*c)/d)*tan(2*a)^2 - 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 2*b*d*x*real_part(cos_int
egral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 4*b*c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) -
4*b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 8*b*c*sin_integral(4*(b*d*x + b*c)/d)
*tan(2*a)*tan(2*b*c/d) - b*c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 + b*c*imag_part(cos_integ
ral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - d*tan(2*b*x)^2*
tan(2*b*c/d)^2 - 2*d*tan(2*b*x)*tan(2*a)*tan(2*b*c/d)^2 - d*tan(2*a)^2*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_in
tegral(4*b*x + 4*b*c/d)) - b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 2*b*d*x*sin_integral(4*(b*d*x + b
*c)/d) + 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c
/d))*tan(2*a) - 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 2*b*c*real_part(cos_integral(-4*
b*x - 4*b*c/d))*tan(2*b*c/d) + b*c*imag_part(cos_integral(4*b*x + 4*b*c/d)) - b*c*imag_part(cos_integral(-4*b*
x - 4*b*c/d)) + 2*b*c*sin_integral(4*(b*d*x + b*c)/d) - d*tan(2*b*x)^2 - 2*d*tan(2*b*x)*tan(2*a) - d*tan(2*a)^
2)/(d^3*x*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + c*d^2*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + d^3*x*tan(2*
b*x)^2*tan(2*a)^2 + d^3*x*tan(2*b*x)^2*tan(2*b*c/d)^2 + d^3*x*tan(2*a)^2*tan(2*b*c/d)^2 + c*d^2*tan(2*b*x)^2*t
an(2*a)^2 + c*d^2*tan(2*b*x)^2*tan(2*b*c/d)^2 + c*d^2*tan(2*a)^2*tan(2*b*c/d)^2 + d^3*x*tan(2*b*x)^2 + d^3*x*t
an(2*a)^2 + d^3*x*tan(2*b*c/d)^2 + c*d^2*tan(2*b*x)^2 + c*d^2*tan(2*a)^2 + c*d^2*tan(2*b*c/d)^2 + d^3*x + c*d^
2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^2,x)

[Out]

int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^2, x)