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

Optimal. Leaf size=158 \[ -\frac{4 b^3 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b c}{d}+4 b x\right )}{3 d^4}-\frac{4 b^3 \cos \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{3 d^4}-\frac{b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac{b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}+\frac{\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac{1}{24 d (c+d x)^3} \]

[Out]

-1/(24*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)

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Rubi [A]  time = 0.227787, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4406, 3297, 3303, 3299, 3302} \[ -\frac{4 b^3 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b c}{d}+4 b x\right )}{3 d^4}-\frac{4 b^3 \cos \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{3 d^4}-\frac{b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac{b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}+\frac{\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac{1}{24 d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(24*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 4406

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]

Rule 3297

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 3303

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 3299

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 3302

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]

Rubi steps

\begin{align*} \int \frac{\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx &=\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 \text{Ci}\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]  time = 1.7543, size = 123, normalized size = 0.78 \[ -\frac{32 b^3 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b (c+d x)}{d}\right )+\frac{d \left (\cos (4 (a+b x)) \left (8 b^2 (c+d x)^2-d^2\right )+d (2 b (c+d x) \sin (4 (a+b x))+d)\right )}{(c+d x)^3}+32 b^3 \cos \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b (c+d x)}{d}\right )}{24 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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])/(24*d^4)

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Maple [A]  time = 0.026, size = 230, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ( -{\frac{{b}^{4}}{32} \left ( -{\frac{4\,\cos \left ( 4\,bx+4\,a \right ) }{3\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{3}d}}-{\frac{4}{3\,d} \left ( -2\,{\frac{\sin \left ( 4\,bx+4\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}+2\,{\frac{1}{d} \left ( -4\,{\frac{\cos \left ( 4\,bx+4\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}-4\,{\frac{1}{d} \left ( 4\,{\frac{1}{d}{\it Si} \left ( 4\,bx+4\,a+4\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 4\,{\frac{-ad+bc}{d}} \right ) }-4\,{\frac{1}{d}{\it Ci} \left ( 4\,bx+4\,a+4\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 4\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \right ) }-{\frac{{b}^{4}}{24\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{3}d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 2.338, size = 346, normalized size = 2.19 \begin{align*} \frac{3 \, b^{4}{\left (E_{4}\left (\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right ) + E_{4}\left (-\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) - b^{4}{\left (3 i \, E_{4}\left (\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right ) - 3 i \, E_{4}\left (-\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + 4*I*(b*x + a)*d - 4*I*a*d)/d) + exp_integral_e(4, -(4*I*b*c + 4*I*(b
*x + a)*d - 4*I*a*d)/d))*cos(-4*(b*c - a*d)/d) - b^4*(3*I*exp_integral_e(4, (4*I*b*c + 4*I*(b*x + a)*d - 4*I*a
*d)/d) - 3*I*exp_integral_e(4, -(4*I*b*c + 4*I*(b*x + a)*d - 4*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)

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Fricas [B]  time = 0.600936, size = 880, normalized size = 5.57 \begin{align*} -\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 )} \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 ) + 2 \,{\left ({\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 ) +{\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 )\right )} \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right )}{3 \,{\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(-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) + 2*((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) + (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))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [C]  time = 1.87002, size = 11486, normalized size = 72.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)