\(\int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx\) [87]

Optimal result

Integrand size = 24, antiderivative size = 158 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {1}{24 d (c+d x)^3}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {4 b^3 \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{3 d^4}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac {4 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4} \]

[Out]

-1/24/d/(d*x+c)^3+1/24*cos(4*b*x+4*a)/d/(d*x+c)^3-1/3*b^2*cos(4*b*x+4*a)/d^3/(d*x+c)-4/3*b^3*cos(4*a-4*b*c/d)*
Si(4*b*c/d+4*b*x)/d^4-4/3*b^3*Ci(4*b*c/d+4*b*x)*sin(4*a-4*b*c/d)/d^4-1/12*b*sin(4*b*x+4*a)/d^2/(d*x+c)^2

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4491, 3378, 3384, 3380, 3383} \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {4 b^3 \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4}-\frac {4 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4}-\frac {b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {1}{24 d (c+d x)^3} \]

[In]

Int[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^4,x]

[Out]

-1/24*1/(d*(c + d*x)^3) + Cos[4*a + 4*b*x]/(24*d*(c + d*x)^3) - (b^2*Cos[4*a + 4*b*x])/(3*d^3*(c + d*x)) - (4*
b^3*CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/(3*d^4) - (b*Sin[4*a + 4*b*x])/(12*d^2*(c + d*x)^2) -
(4*b^3*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(3*d^4)

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)^4}-\frac {\cos (4 a+4 b x)}{8 (c+d x)^4}\right ) \, dx \\ & = -\frac {1}{24 d (c+d x)^3}-\frac {1}{8} \int \frac {\cos (4 a+4 b x)}{(c+d x)^4} \, dx \\ & = -\frac {1}{24 d (c+d x)^3}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}+\frac {b \int \frac {\sin (4 a+4 b x)}{(c+d x)^3} \, dx}{6 d} \\ & = -\frac {1}{24 d (c+d x)^3}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}+\frac {b^2 \int \frac {\cos (4 a+4 b x)}{(c+d x)^2} \, dx}{3 d^2} \\ & = -\frac {1}{24 d (c+d x)^3}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac {\left (4 b^3\right ) \int \frac {\sin (4 a+4 b x)}{c+d x} \, dx}{3 d^3} \\ & = -\frac {1}{24 d (c+d x)^3}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac {\left (4 b^3 \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}{3 d^3}-\frac {\left (4 b^3 \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}{3 d^3} \\ & = -\frac {1}{24 d (c+d x)^3}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {4 b^3 \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{3 d^4}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac {4 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {32 b^3 \operatorname {CosIntegral}\left (\frac {4 b (c+d x)}{d}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )+\frac {d \left (\left (-d^2+8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))+d (d+2 b (c+d x) \sin (4 (a+b x)))\right )}{(c+d x)^3}+32 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )}{24 d^4} \]

[In]

Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^4,x]

[Out]

-1/24*(32*b^3*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + (d*((-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a +
b*x)] + d*(d + 2*b*(c + d*x)*Sin[4*(a + b*x)])))/(c + d*x)^3 + 32*b^3*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c
+ d*x))/d])/d^4

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(\frac {-\frac {b^{4} \left (-\frac {4 \cos \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}-\frac {4 \left (-\frac {2 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {8 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {8 \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{3 d}\right )}{32}-\frac {b^{4}}{24 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}}{b}\)	\(230\)
default	\(\frac {-\frac {b^{4} \left (-\frac {4 \cos \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}-\frac {4 \left (-\frac {2 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {8 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {8 \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{3 d}\right )}{32}-\frac {b^{4}}{24 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}}{b}\)	\(230\)
risch	\(-\frac {1}{24 d \left (d x +c \right )^{3}}+\frac {2 i b^{3} {\mathrm e}^{-\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (4 i b x +4 i a -\frac {4 i \left (a d -c b \right )}{d}\right )}{3 d^{4}}-\frac {2 i b^{3} {\mathrm e}^{\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-4 i b x -4 i a -\frac {4 \left (-i a d +i c b \right )}{d}\right )}{3 d^{4}}+\frac {\left (-16 b^{5} d^{5} x^{5}-80 b^{5} c \,d^{4} x^{4}-160 b^{5} c^{2} d^{3} x^{3}-160 b^{5} c^{3} d^{2} x^{2}-80 b^{5} c^{4} d x +2 b^{3} d^{5} x^{3}-16 b^{5} c^{5}+6 b^{3} c \,d^{4} x^{2}+6 b^{3} c^{2} d^{3} x +2 b^{3} c^{3} d^{2}\right ) \cos \left (4 x b +4 a \right )}{48 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}+\frac {i \left (4 i b^{4} d^{5} x^{4}+16 i b^{4} c \,d^{4} x^{3}+24 i b^{4} c^{2} d^{3} x^{2}+16 i b^{4} c^{3} d^{2} x +4 i b^{4} c^{4} d \right ) \sin \left (4 x b +4 a \right )}{48 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}\)	\(423\)

[In]

int(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^4,x,method=_RETURNVERBOSE)

[Out]

1/b*(-1/32*b^4*(-4/3*cos(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))^3/d-4/3*(-2*sin(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))^2/d+2
*(-4*cos(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))/d-4*(-4*Si(-4*x*b-4*a-4*(-a*d+b*c)/d)*cos(4*(-a*d+b*c)/d)/d-4*Ci(4*x*
b+4*a+4*(-a*d+b*c)/d)*sin(4*(-a*d+b*c)/d)/d)/d)/d)/d)-1/24*b^4/(-a*d+c*b+d*(b*x+a))^3/d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (146) = 292\).

Time = 0.27 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.21 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{4} - {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (2 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{3} - {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{3 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^4,x, algorithm="fricas")

[Out]

-1/3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d + (8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - d^3)*cos(b*x +
a)^4 - (8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - d^3)*cos(b*x + a)^2 + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2
+ 3*b^3*c^2*d*x + b^3*c^3)*cos_integral(4*(b*d*x + b*c)/d)*sin(-4*(b*c - a*d)/d) + 4*(b^3*d^3*x^3 + 3*b^3*c*d
^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos(-4*(b*c - a*d)/d)*sin_integral(4*(b*d*x + b*c)/d) + (2*(b*d^3*x + b*c*d^
2)*cos(b*x + a)^3 - (b*d^3*x + b*c*d^2)*cos(b*x + a))*sin(b*x + a))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3
*d^4)

Sympy [F]

\[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]

[In]

integrate(cos(b*x+a)**2*sin(b*x+a)**2/(d*x+c)**4,x)

[Out]

Integral(sin(a + b*x)**2*cos(a + b*x)**2/(c + d*x)**4, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\frac {3 \, b^{4} {\left (E_{4}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{4} {\left (i \, E_{4}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{4}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, b^{4}}{48 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^4,x, algorithm="maxima")

[Out]

1/48*(3*b^4*(exp_integral_e(4, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(4, -4*(-I*b*c - I*(b*x +
a)*d + I*a*d)/d))*cos(-4*(b*c - a*d)/d) + 3*b^4*(I*exp_integral_e(4, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) -
I*exp_integral_e(4, -4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-4*(b*c - a*d)/d) - 2*b^4)/((b^3*c^3*d - 3*a*b
^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*
a*b*c*d^3 + a^2*d^4)*(b*x + a))*b)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.57 (sec) , antiderivative size = 8508, normalized size of antiderivative = 53.85 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^4,x, algorithm="giac")

[Out]

-1/12*(8*b^3*d^3*x^3*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 - 8*b^3*d
^3*x^3*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 + 16*b^3*d^3*x^3*sin_i
ntegral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 16*b^3*d^3*x^3*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) + 16*b^3*d^3*x^3*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) - 16*b^3*d^3*x^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*ta
n(2*a)*tan(2*b*c/d)^2 - 16*b^3*d^3*x^3*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 + 24*b^3*c*d^2*x^2*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
4*b^3*c*d^2*x^2*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 + 48*b^3*c*d^
2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^3*d^3*x^3*imag_part(cos_int
egral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + 8*b^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(
2*b*x)^2*tan(2*a)^2 - 16*b^3*d^3*x^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2 + 32*b^3*d^3*x^3*
imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 32*b^3*d^3*x^3*imag_part(cos_int
egral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 64*b^3*d^3*x^3*sin_integral(4*(b*d*x + b*c)/d)*t
an(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 48*b^3*c*d^2*x^2*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) + 48*b^3*c*d^2*x^2*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) - 8*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + 8*b^3*d^3*x^3*
imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 - 16*b^3*d^3*x^3*sin_integral(4*(b*d*x +
b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d)^2 - 48*b^3*c*d^2*x^2*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 - 48*b^3*c*d^2*x^2*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 + 8*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^3*d^3*x^3
*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 16*b^3*d^3*x^3*sin_integral(4*(b*d*x +
b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2 + 24*b^3*c^2*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 - 24*b^3*c^2*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 + 48*b^3*c^2*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 + 16*b^3*d^3*
x^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a) + 16*b^3*d^3*x^3*real_part(cos_integral(-4*
b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 24*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2
*tan(2*a)^2 + 24*b^3*c*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 - 48*b^3*c*d^
2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2 - 16*b^3*d^3*x^3*real_part(cos_integral(4*b*x +
4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 16*b^3*d^3*x^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*ta
n(2*b*c/d) + 96*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 96
*b^3*c*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 192*b^3*c*d^2*x^
2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 16*b^3*d^3*x^3*real_part(cos_integral(4
*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 16*b^3*d^3*x^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2
*tan(2*b*c/d) + 48*b^3*c^2*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) +
48*b^3*c^2*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) - 24*b^3*c*d^2*
x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + 24*b^3*c*d^2*x^2*imag_part(cos_inte
gral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 - 48*b^3*c*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b
*x)^2*tan(2*b*c/d)^2 - 16*b^3*d^3*x^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 16*b^
3*d^3*x^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 48*b^3*c^2*d*x*real_part(cos_int
egral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 48*b^3*c^2*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 + 24*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan
(2*a)^2*tan(2*b*c/d)^2 - 24*b^3*c*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2
+ 48*b^3*c*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*d^3*x^2*tan(2*b*x)^2*tan(
2*a)^2*tan(2*b*c/d)^2 + 8*b^3*c^3*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 - 8*b^3*c^3*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 + 16*b^3*c^3*
sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^3*d^3*x^3*imag_part(cos_integral(
4*b*x + 4*b*c/d))*tan(2*b*x)^2 - 8*b^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 + 16*b^3
*d^3*x^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2 + 48*b^3*c*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/
d))*tan(2*b*x)^2*tan(2*a) + 48*b^3*c*d^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) -
8*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2 + 8*b^3*d^3*x^3*imag_part(cos_integral(-4*b
*x - 4*b*c/d))*tan(2*a)^2 - 16*b^3*d^3*x^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2 - 24*b^3*c^2*d*x*imag_pa
rt(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + 24*b^3*c^2*d*x*imag_part(cos_integral(-4*b*x - 4*b
*c/d))*tan(2*b*x)^2*tan(2*a)^2 - 48*b^3*c^2*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2 - 48*b
^3*c*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 48*b^3*c*d^2*x^2*real_part(c
os_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 32*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c
/d))*tan(2*a)*tan(2*b*c/d) - 32*b^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) +
64*b^3*d^3*x^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d) + 96*b^3*c^2*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) - 96*b^3*c^2*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) + 192*b^3*c^2*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*ta
n(2*b*c/d) + 48*b^3*c*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 48*b^3*c*d^2*
x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 16*b^3*c^3*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) + 16*b^3*c^3*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) - 8*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 + 8*b
^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 16*b^3*d^3*x^3*sin_integral(4*(b*d*x + b
*c)/d)*tan(2*b*c/d)^2 - 24*b^3*c^2*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 +
24*b^3*c^2*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 - 48*b^3*c^2*d*x*sin_inte
gral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d)^2 - 48*b^3*c*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d)
)*tan(2*a)*tan(2*b*c/d)^2 - 48*b^3*c*d^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2
- 16*b^3*c^3*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 - 16*b^3*c^3*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 + 24*b^3*c^2*d*x*imag_part(cos_integ
ral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - 24*b^3*c^2*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*a)^2*tan(2*b*c/d)^2 + 48*b^3*c^2*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^2*c*d^
2*x*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 24*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b
*x)^2 - 24*b^3*c*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 + 48*b^3*c*d^2*x^2*sin_integra
l(4*(b*d*x + b*c)/d)*tan(2*b*x)^2 + 16*b^3*d^3*x^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 16*b^3*
d^3*x^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) + 48*b^3*c^2*d*x*real_part(cos_integral(4*b*x + 4*b
*c/d))*tan(2*b*x)^2*tan(2*a) + 48*b^3*c^2*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)
- 24*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2 + 24*b^3*c*d^2*x^2*imag_part(cos_integr
al(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 48*b^3*c*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2 + 4*b^2*d^3*x^2
*tan(2*b*x)^2*tan(2*a)^2 - 8*b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + 8*b^3*
c^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 - 16*b^3*c^3*sin_integral(4*(b*d*x + b*c
)/d)*tan(2*b*x)^2*tan(2*a)^2 - 16*b^3*d^3*x^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 16*b^3*d
^3*x^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) - 48*b^3*c^2*d*x*real_part(cos_integral(4*b*x +
4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 48*b^3*c^2*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*ta
n(2*b*c/d) + 96*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) - 96*b^3*c*d^2*x^
2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 192*b^3*c*d^2*x^2*sin_integral(4*(b*d*x +
b*c)/d)*tan(2*a)*tan(2*b*c/d) + 32*b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(
2*b*c/d) - 32*b^3*c^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 64*b^3*c^
3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 48*b^3*c^2*d*x*real_part(cos_integral(4
*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 48*b^3*c^2*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2
*tan(2*b*c/d) - 24*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 + 24*b^3*c*d^2*x^2*im
ag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 48*b^3*c*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(
2*b*c/d)^2 - 4*b^2*d^3*x^2*tan(2*b*x)^2*tan(2*b*c/d)^2 - 8*b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*ta
n(2*b*x)^2*tan(2*b*c/d)^2 + 8*b^3*c^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 -
16*b^3*c^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d)^2 - 48*b^3*c^2*d*x*real_part(cos_integral
(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 48*b^3*c^2*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)
*tan(2*b*c/d)^2 - 16*b^2*d^3*x^2*tan(2*b*x)*tan(2*a)*tan(2*b*c/d)^2 - 4*b^2*d^3*x^2*tan(2*a)^2*tan(2*b*c/d)^2
+ 8*b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^3*c^3*imag_part(cos_integ
ral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 16*b^3*c^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2
*b*c/d)^2 + 4*b^2*c^2*d*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^3*d^3*x^3*imag_part(cos_integral(4*b*x +
4*b*c/d)) - 8*b^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 16*b^3*d^3*x^3*sin_integral(4*(b*d*x + b
*c)/d) + 24*b^3*c^2*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 - 24*b^3*c^2*d*x*imag_part(cos_i
ntegral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 + 48*b^3*c^2*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2 + 48*b^3
*c*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 48*b^3*c*d^2*x^2*real_part(cos_integral(-4*b*x
- 4*b*c/d))*tan(2*a) + 16*b^3*c^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a) + 16*b^3*c^3*
real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 24*b^3*c^2*d*x*imag_part(cos_integral(4*b*x
+ 4*b*c/d))*tan(2*a)^2 + 24*b^3*c^2*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 48*b^3*c^2*d*x*
sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2 + 8*b^2*c*d^2*x*tan(2*b*x)^2*tan(2*a)^2 - 48*b^3*c*d^2*x^2*real_par
t(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 48*b^3*c*d^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*b*c/d) - 16*b^3*c^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 16*b^3*c^3*real_pa
rt(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 96*b^3*c^2*d*x*imag_part(cos_integral(4*b*x + 4
*b*c/d))*tan(2*a)*tan(2*b*c/d) - 96*b^3*c^2*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d
) + 192*b^3*c^2*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d) + 16*b^3*c^3*real_part(cos_integral(
4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 16*b^3*c^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*ta
n(2*b*c/d) - 24*b^3*c^2*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 + 24*b^3*c^2*d*x*imag_part
(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 48*b^3*c^2*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^
2 - 8*b^2*c*d^2*x*tan(2*b*x)^2*tan(2*b*c/d)^2 - 16*b^3*c^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*t
an(2*b*c/d)^2 - 16*b^3*c^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 32*b^2*c*d^2*x*
tan(2*b*x)*tan(2*a)*tan(2*b*c/d)^2 - 2*b*d^3*x*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 8*b^2*c*d^2*x*tan(2*a)^2
*tan(2*b*c/d)^2 - 2*b*d^3*x*tan(2*b*x)*tan(2*a)^2*tan(2*b*c/d)^2 + 24*b^3*c*d^2*x^2*imag_part(cos_integral(4*b
*x + 4*b*c/d)) - 24*b^3*c*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 48*b^3*c*d^2*x^2*sin_integral(4*
(b*d*x + b*c)/d) - 4*b^2*d^3*x^2*tan(2*b*x)^2 + 8*b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^
2 - 8*b^3*c^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 + 16*b^3*c^3*sin_integral(4*(b*d*x + b*c)
/d)*tan(2*b*x)^2 + 48*b^3*c^2*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 48*b^3*c^2*d*x*real_part
(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - 16*b^2*d^3*x^2*tan(2*b*x)*tan(2*a) - 4*b^2*d^3*x^2*tan(2*a)^2 - 8*
b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2 + 8*b^3*c^3*imag_part(cos_integral(-4*b*x - 4*b*c/
d))*tan(2*a)^2 - 16*b^3*c^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2 + 4*b^2*c^2*d*tan(2*b*x)^2*tan(2*a)^2 -
48*b^3*c^2*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 48*b^3*c^2*d*x*real_part(cos_integral(
-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 32*b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) -
32*b^3*c^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 64*b^3*c^3*sin_integral(4*(b*d*x
+ b*c)/d)*tan(2*a)*tan(2*b*c/d) + 4*b^2*d^3*x^2*tan(2*b*c/d)^2 - 8*b^3*c^3*imag_part(cos_integral(4*b*x + 4*b*
c/d))*tan(2*b*c/d)^2 + 8*b^3*c^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 16*b^3*c^3*sin_int
egral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - 4*b^2*c^2*d*tan(2*b*x)^2*tan(2*b*c/d)^2 - 16*b^2*c^2*d*tan(2*b*x)*ta
n(2*a)*tan(2*b*c/d)^2 - 2*b*c*d^2*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 4*b^2*c^2*d*tan(2*a)^2*tan(2*b*c/d)^2
- 2*b*c*d^2*tan(2*b*x)*tan(2*a)^2*tan(2*b*c/d)^2 + 24*b^3*c^2*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d)) -
24*b^3*c^2*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 48*b^3*c^2*d*x*sin_integral(4*(b*d*x + b*c)/d) - 8*
b^2*c*d^2*x*tan(2*b*x)^2 + 16*b^3*c^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 16*b^3*c^3*real_part
(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - 32*b^2*c*d^2*x*tan(2*b*x)*tan(2*a) - 2*b*d^3*x*tan(2*b*x)^2*tan(2*
a) - 8*b^2*c*d^2*x*tan(2*a)^2 - 2*b*d^3*x*tan(2*b*x)*tan(2*a)^2 - 16*b^3*c^3*real_part(cos_integral(4*b*x + 4*
b*c/d))*tan(2*b*c/d) - 16*b^3*c^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 8*b^2*c*d^2*x*tan(2
*b*c/d)^2 + 2*b*d^3*x*tan(2*b*x)*tan(2*b*c/d)^2 + 2*b*d^3*x*tan(2*a)*tan(2*b*c/d)^2 + 4*b^2*d^3*x^2 + 8*b^3*c^
3*imag_part(cos_integral(4*b*x + 4*b*c/d)) - 8*b^3*c^3*imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 16*b^3*c^3*
sin_integral(4*(b*d*x + b*c)/d) - 4*b^2*c^2*d*tan(2*b*x)^2 - 16*b^2*c^2*d*tan(2*b*x)*tan(2*a) - 2*b*c*d^2*tan(
2*b*x)^2*tan(2*a) - 4*b^2*c^2*d*tan(2*a)^2 - 2*b*c*d^2*tan(2*b*x)*tan(2*a)^2 + 4*b^2*c^2*d*tan(2*b*c/d)^2 + 2*
b*c*d^2*tan(2*b*x)*tan(2*b*c/d)^2 + d^3*tan(2*b*x)^2*tan(2*b*c/d)^2 + 2*b*c*d^2*tan(2*a)*tan(2*b*c/d)^2 + 2*d^
3*tan(2*b*x)*tan(2*a)*tan(2*b*c/d)^2 + d^3*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^2*c*d^2*x + 2*b*d^3*x*tan(2*b*x) +
2*b*d^3*x*tan(2*a) + 4*b^2*c^2*d + 2*b*c*d^2*tan(2*b*x) + d^3*tan(2*b*x)^2 + 2*b*c*d^2*tan(2*a) + 2*d^3*tan(2*
b*x)*tan(2*a) + d^3*tan(2*a)^2)/(d^7*x^3*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 3*c*d^6*x^2*tan(2*b*x)^2*tan
(2*a)^2*tan(2*b*c/d)^2 + d^7*x^3*tan(2*b*x)^2*tan(2*a)^2 + d^7*x^3*tan(2*b*x)^2*tan(2*b*c/d)^2 + d^7*x^3*tan(2
*a)^2*tan(2*b*c/d)^2 + 3*c^2*d^5*x*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 3*c*d^6*x^2*tan(2*b*x)^2*tan(2*a)^
2 + 3*c*d^6*x^2*tan(2*b*x)^2*tan(2*b*c/d)^2 + 3*c*d^6*x^2*tan(2*a)^2*tan(2*b*c/d)^2 + c^3*d^4*tan(2*b*x)^2*tan
(2*a)^2*tan(2*b*c/d)^2 + d^7*x^3*tan(2*b*x)^2 + d^7*x^3*tan(2*a)^2 + 3*c^2*d^5*x*tan(2*b*x)^2*tan(2*a)^2 + d^7
*x^3*tan(2*b*c/d)^2 + 3*c^2*d^5*x*tan(2*b*x)^2*tan(2*b*c/d)^2 + 3*c^2*d^5*x*tan(2*a)^2*tan(2*b*c/d)^2 + 3*c*d^
6*x^2*tan(2*b*x)^2 + 3*c*d^6*x^2*tan(2*a)^2 + c^3*d^4*tan(2*b*x)^2*tan(2*a)^2 + 3*c*d^6*x^2*tan(2*b*c/d)^2 + c
^3*d^4*tan(2*b*x)^2*tan(2*b*c/d)^2 + c^3*d^4*tan(2*a)^2*tan(2*b*c/d)^2 + d^7*x^3 + 3*c^2*d^5*x*tan(2*b*x)^2 +
3*c^2*d^5*x*tan(2*a)^2 + 3*c^2*d^5*x*tan(2*b*c/d)^2 + 3*c*d^6*x^2 + c^3*d^4*tan(2*b*x)^2 + c^3*d^4*tan(2*a)^2
+ c^3*d^4*tan(2*b*c/d)^2 + 3*c^2*d^5*x + c^3*d^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^4} \,d x \]

[In]

int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^4,x)

[Out]

int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^4, x)