3.374 \(\int \frac{\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx\)

Optimal. Leaf size=136 \[ -\frac{4 b^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}+\frac{4 b^2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}+\frac{4 b \sin (a+b x) \cos (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2} \]

[Out]

(-3*Cos[a + b*x]^2)/(2*d*(c + d*x)^2) - (4*b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^3 + (4*b
*Cos[a + b*x]*Sin[a + b*x])/(d^2*(c + d*x)) + Sin[a + b*x]^2/(2*d*(c + d*x)^2) + (4*b^2*Sin[2*a - (2*b*c)/d]*S
inIntegral[(2*b*c)/d + 2*b*x])/d^3

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Rubi [A]  time = 0.374217, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4431, 3314, 31, 3312, 3303, 3299, 3302} \[ -\frac{4 b^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}+\frac{4 b^2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}+\frac{4 b \sin (a+b x) \cos (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^3,x]

[Out]

(-3*Cos[a + b*x]^2)/(2*d*(c + d*x)^2) - (4*b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^3 + (4*b
*Cos[a + b*x]*Sin[a + b*x])/(d^2*(c + d*x)) + Sin[a + b*x]^2/(2*d*(c + d*x)^2) + (4*b^2*Sin[2*a - (2*b*c)/d]*S
inIntegral[(2*b*c)/d + 2*b*x])/d^3

Rule 4431

Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && M
emberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d,
1]

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && 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{\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx &=\int \left (\frac{3 \cos ^2(a+b x)}{(c+d x)^3}-\frac{\sin ^2(a+b x)}{(c+d x)^3}\right ) \, dx\\ &=3 \int \frac{\cos ^2(a+b x)}{(c+d x)^3} \, dx-\int \frac{\sin ^2(a+b x)}{(c+d x)^3} \, dx\\ &=-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac{4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac{b^2 \int \frac{1}{c+d x} \, dx}{d^2}+\frac{\left (2 b^2\right ) \int \frac{\sin ^2(a+b x)}{c+d x} \, dx}{d^2}+\frac{\left (3 b^2\right ) \int \frac{1}{c+d x} \, dx}{d^2}-\frac{\left (6 b^2\right ) \int \frac{\cos ^2(a+b x)}{c+d x} \, dx}{d^2}\\ &=-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac{2 b^2 \log (c+d x)}{d^3}+\frac{4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}+\frac{\left (2 b^2\right ) \int \left (\frac{1}{2 (c+d x)}-\frac{\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx}{d^2}-\frac{\left (6 b^2\right ) \int \left (\frac{1}{2 (c+d x)}+\frac{\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac{4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac{b^2 \int \frac{\cos (2 a+2 b x)}{c+d x} \, dx}{d^2}-\frac{\left (3 b^2\right ) \int \frac{\cos (2 a+2 b x)}{c+d x} \, dx}{d^2}\\ &=-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2}+\frac{4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac{\left (b^2 \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^2}-\frac{\left (3 b^2 \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^2}+\frac{\left (b^2 \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^2}+\frac{\left (3 b^2 \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^2}\\ &=-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2}-\frac{4 b^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Ci}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}+\frac{4 b \cos (a+b x) \sin (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}+\frac{4 b^2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.998589, size = 104, normalized size = 0.76 \[ -\frac{8 b^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )-8 b^2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+\frac{d (-4 b (c+d x) \sin (2 (a+b x))+2 d \cos (2 (a+b x))+d)}{(c+d x)^2}}{2 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^3,x]

[Out]

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

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Maple [A]  time = 0.038, size = 207, normalized size = 1.5 \begin{align*}{\frac{1}{2\,d \left ( dx+c \right ) ^{2}}}+4\,{\frac{1}{b} \left ( 1/4\,{b}^{3} \left ( -{\frac{\cos \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}-{\frac{1}{d} \left ( -2\,{\frac{\sin \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 2\,{\frac{-ad+bc}{d}} \right ) }+2\,{\frac{1}{d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 2\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) -1/4\,{\frac{{b}^{3}}{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^3,x)

[Out]

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

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Maxima [C]  time = 1.4501, size = 176, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (E_{3}\left (\frac{2 i \, b d x + 2 i \, b c}{d}\right ) + E_{3}\left (-\frac{2 i \, b d x + 2 i \, b c}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) -{\left (2 i \, E_{3}\left (\frac{2 i \, b d x + 2 i \, b c}{d}\right ) - 2 i \, E_{3}\left (-\frac{2 i \, b d x + 2 i \, b c}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + 1}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.530617, size = 516, normalized size = 3.79 \begin{align*} -\frac{4 \, d^{2} \cos \left (b x + a\right )^{2} - 8 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 8 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\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^{2} + 4 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname{Ci}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname{Ci}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)**3,x)

[Out]

Timed out

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Giac [C]  time = 2.1155, size = 12712, normalized size = 93.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^3,x, algorithm="giac")

[Out]

1/2*(4*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b^2*d^2*x
^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 - 8*b^2*d^2*x^2*imag_part(co
s_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x
- 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) - 16*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan
(1/2*a)^4*tan(b*c/d) + 16*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c
/d)^2 - 16*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 32*b^2
*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_inte
gral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*
c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 - 4*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2
*tan(1/2*a)^4 - 4*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4 + 32*b^2*d^2*x
^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) + 32*b^2*d^2*x^2*real_part(cos_
integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2
*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) + 16*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^
2*tan(1/2*a)^4*tan(b*c/d) - 32*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) -
24*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 - 24*b^2*d^2*x^2*
real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + 32*b^2*c*d*x*imag_part(cos_in
tegral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 - 32*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2
*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 64*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/
2*a)^3*tan(b*c/d)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b^2
*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b^2*c^2*real_part(cos_integra
l(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))
*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 - 16*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan
(1/2*a)^3 + 16*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3 - 32*b^2*d^2*x^2*
sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^3 - 8*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))
*tan(b*x)^2*tan(1/2*a)^4 - 8*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4 + 48*
b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) - 48*b^2*d^2*x^2*imag_
part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) + 96*b^2*d^2*x^2*sin_integral(2*(b*d*x
+ b*c)/d)*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) + 64*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)
^2*tan(1/2*a)^3*tan(b*c/d) + 64*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*ta
n(b*c/d) - 8*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d) + 8*b^2*d^2*x^2*imag
_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d) - 16*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)
*tan(1/2*a)^4*tan(b*c/d) - 8*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/
d) + 8*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) - 16*b^2*c^2*sin_i
ntegral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) - 16*b^2*d^2*x^2*imag_part(cos_integral(2*b*x +
2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 + 16*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*
x)^2*tan(1/2*a)*tan(b*c/d)^2 - 32*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)
^2 - 48*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 - 48*b^2*c*d*x
*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + 16*b^2*d^2*x^2*imag_part(cos
_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d)^2 - 16*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d
))*tan(1/2*a)^3*tan(b*c/d)^2 + 32*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^3*tan(b*c/d)^2 + 16*b
^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 - 16*b^2*c^2*imag_part(co
s_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 32*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d
)*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4*tan
(b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2 + 24*b^2*d^2*x^2*r
eal_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 + 24*b^2*d^2*x^2*real_part(cos_integral(-2*b*x
- 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 - 32*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/
2*a)^3 + 32*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3 - 64*b^2*c*d*x*sin_int
egral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^3 - 4*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(
1/2*a)^4 - 4*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4 - 4*b^2*c^2*real_part(cos_inte
gral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4 - 4*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^
2*tan(1/2*a)^4 - 32*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) - 32
*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) + 96*b^2*c*d*x*imag_pa
rt(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) - 96*b^2*c*d*x*imag_part(cos_integral(-2*
b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) + 192*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*
tan(1/2*a)^2*tan(b*c/d) + 32*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) + 32
*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) + 32*b^2*c^2*real_part(cos_inte
gral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) + 32*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d
))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4*tan
(b*c/d) + 16*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d) - 32*b^2*c*d*x*sin_in
tegral(2*(b*d*x + b*c)/d)*tan(1/2*a)^4*tan(b*c/d) + 4*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan
(b*x)^2*tan(b*c/d)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 32*b^
2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 + 32*b^2*c*d*x*imag_part(c
os_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 - 64*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/
d)*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 - 24*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2*t
an(b*c/d)^2 - 24*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)^2 - 24*b^2*c^2*
real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 - 24*b^2*c^2*real_part(cos_integ
ral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + 32*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*
c/d))*tan(1/2*a)^3*tan(b*c/d)^2 - 32*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/
d)^2 + 64*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^3*tan(b*c/d)^2 + 16*b*d^2*x*tan(b*x)^2*tan(1/2*
a)^3*tan(b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b^2*c^2*r
eal_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2 + 8*b*d^2*x*tan(b*x)*tan(1/2*a)^4*tan(b*c/d
)^2 + 16*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a) - 16*b^2*d^2*x^2*imag_part
(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a) + 32*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*
x)^2*tan(1/2*a) + 48*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 + 48*b^2*c*d*x
*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 - 16*b^2*d^2*x^2*imag_part(cos_integral(2*b
*x + 2*b*c/d))*tan(1/2*a)^3 + 16*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3 - 32*b^2*d
^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^3 - 16*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(
b*x)^2*tan(1/2*a)^3 + 16*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3 - 32*b^2*c^
2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^3 - 8*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d
))*tan(1/2*a)^4 - 8*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4 - 8*b^2*d^2*x^2*imag_part
(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(b*x)^2*tan(b*c/d) - 16*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d) - 64*b^2*c*d*x*
real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) - 64*b^2*c*d*x*real_part(cos_integra
l(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) + 48*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d)
)*tan(1/2*a)^2*tan(b*c/d) - 48*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) +
96*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^2*tan(b*c/d) + 48*b^2*c^2*imag_part(cos_integral(2*
b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) - 48*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(
b*x)^2*tan(1/2*a)^2*tan(b*c/d) + 96*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)
+ 64*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) + 64*b^2*c*d*x*real_part(cos_
integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) - 8*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1
/2*a)^4*tan(b*c/d) + 8*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d) - 16*b^2*c^2*
sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^4*tan(b*c/d) + 8*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))
*tan(b*x)^2*tan(b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 16*
b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/d)^2 + 16*b^2*d^2*x^2*imag_part(cos_in
tegral(-2*b*x - 2*b*c/d))*tan(1/2*a)*tan(b*c/d)^2 - 32*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)*
tan(b*c/d)^2 - 16*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 + 16*b^2
*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 - 32*b^2*c^2*sin_integral(2*
(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 - 48*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*ta
n(1/2*a)^2*tan(b*c/d)^2 - 48*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)^2 + 1
6*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d)^2 - 16*b^2*c^2*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d)^2 + 32*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^3*tan(
b*c/d)^2 + 16*b*c*d*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 8*b*c*d*tan(b*x)*tan(1/2*a)^4*tan(b*c/d)^2 + d^2*ta
n(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 - 4*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 - 4*b^2
*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 32*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*
b*c/d))*tan(b*x)^2*tan(1/2*a) - 32*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a) +
64*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a) + 24*b^2*d^2*x^2*real_part(cos_integral(2*
b*x + 2*b*c/d))*tan(1/2*a)^2 + 24*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2 + 24*b^2*
c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 + 24*b^2*c^2*real_part(cos_integral(-2*b*
x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 - 32*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3 + 3
2*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3 - 64*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)
/d)*tan(1/2*a)^3 + 16*b*d^2*x*tan(b*x)^2*tan(1/2*a)^3 - 4*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan
(1/2*a)^4 - 4*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4 + 8*b*d^2*x*tan(b*x)*tan(1/2*a)^4
- 16*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 16*b^2*c*d*x*imag_part(cos_in
tegral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 32*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(
b*c/d) - 32*b^2*d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/d) - 32*b^2*d^2*x^2*real_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)*tan(b*c/d) - 32*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d)
)*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) - 32*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a
)*tan(b*c/d) + 96*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) - 96*b^2*c*d*x*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) + 192*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d
)*tan(1/2*a)^2*tan(b*c/d) + 32*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) + 32*b
^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) + 4*b^2*d^2*x^2*real_part(cos_integra
l(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 + 4*b^
2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(-2*b
*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 32*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(
b*c/d)^2 + 32*b^2*c*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)*tan(b*c/d)^2 - 64*b^2*c*d*x*sin_i
ntegral(2*(b*d*x + b*c)/d)*tan(1/2*a)*tan(b*c/d)^2 - 16*b*d^2*x*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 - 24*b^2*c^
2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)^2 - 24*b^2*c^2*real_part(cos_integral(-2*b*
x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)^2 - 48*b*d^2*x*tan(b*x)*tan(1/2*a)^2*tan(b*c/d)^2 - 16*b*d^2*x*tan(1/2*a
)^3*tan(b*c/d)^2 - 8*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 - 8*b^2*c*d*x*real_part(cos
_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 + 16*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a) -
16*b^2*d^2*x^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a) + 32*b^2*d^2*x^2*sin_integral(2*(b*d*x +
b*c)/d)*tan(1/2*a) + 16*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a) - 16*b^2*c^2*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a) + 32*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan
(b*x)^2*tan(1/2*a) + 48*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2 + 48*b^2*c*d*x*real_pa
rt(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2 - 16*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*
a)^3 + 16*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3 - 32*b^2*c^2*sin_integral(2*(b*d*x +
b*c)/d)*tan(1/2*a)^3 + 16*b*c*d*tan(b*x)^2*tan(1/2*a)^3 + 8*b*c*d*tan(b*x)*tan(1/2*a)^4 + d^2*tan(b*x)^2*tan(1
/2*a)^4 - 8*b^2*d^2*x^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_inte
gral(-2*b*x - 2*b*c/d))*tan(b*c/d) - 16*b^2*d^2*x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d) - 8*b^2*c^2*ima
g_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 8*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/
d))*tan(b*x)^2*tan(b*c/d) - 16*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d) - 64*b^2*c*d*x*re
al_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/d) - 64*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*
b*c/d))*tan(1/2*a)*tan(b*c/d) + 48*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) -
48*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) + 96*b^2*c^2*sin_integral(2*(b*d*
x + b*c)/d)*tan(1/2*a)^2*tan(b*c/d) + 8*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + 8*b^
2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - 16*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b
*c/d))*tan(1/2*a)*tan(b*c/d)^2 + 16*b^2*c^2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)*tan(b*c/d)^2
- 32*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)*tan(b*c/d)^2 - 16*b*c*d*tan(b*x)^2*tan(1/2*a)*tan(b*c/
d)^2 - 48*b*c*d*tan(b*x)*tan(1/2*a)^2*tan(b*c/d)^2 - 14*d^2*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 - 16*b*c*d*ta
n(1/2*a)^3*tan(b*c/d)^2 - 16*d^2*tan(b*x)*tan(1/2*a)^3*tan(b*c/d)^2 - 3*d^2*tan(1/2*a)^4*tan(b*c/d)^2 - 4*b^2*
d^2*x^2*real_part(cos_integral(2*b*x + 2*b*c/d)) - 4*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d)) - 4
*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 - 4*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c
/d))*tan(b*x)^2 + 32*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a) - 32*b^2*c*d*x*imag_part(co
s_integral(-2*b*x - 2*b*c/d))*tan(1/2*a) + 64*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a) - 16*b*d^2*
x*tan(b*x)^2*tan(1/2*a) + 24*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2 + 24*b^2*c^2*real_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2 - 48*b*d^2*x*tan(b*x)*tan(1/2*a)^2 - 16*b*d^2*x*tan(1/2*a)^3
- 16*b^2*c*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 16*b^2*c*d*x*imag_part(cos_integral(-2*b*
x - 2*b*c/d))*tan(b*c/d) - 32*b^2*c*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d) - 32*b^2*c^2*real_part(cos_
integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/d) - 32*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/
2*a)*tan(b*c/d) + 4*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 + 4*b^2*c^2*real_part(cos_in
tegral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 + 8*b*d^2*x*tan(b*x)*tan(b*c/d)^2 + 16*b*d^2*x*tan(1/2*a)*tan(b*c/d)^2
- 8*b^2*c*d*x*real_part(cos_integral(2*b*x + 2*b*c/d)) - 8*b^2*c*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))
+ 16*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a) - 16*b^2*c^2*imag_part(cos_integral(-2*b*x -
2*b*c/d))*tan(1/2*a) + 32*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a) - 16*b*c*d*tan(b*x)^2*tan(1/2*a)
- 48*b*c*d*tan(b*x)*tan(1/2*a)^2 - 14*d^2*tan(b*x)^2*tan(1/2*a)^2 - 16*b*c*d*tan(1/2*a)^3 - 16*d^2*tan(b*x)*t
an(1/2*a)^3 - 3*d^2*tan(1/2*a)^4 - 8*b^2*c^2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 8*b^2*c^2*i
mag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) - 16*b^2*c^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*c/d) +
8*b*c*d*tan(b*x)*tan(b*c/d)^2 + d^2*tan(b*x)^2*tan(b*c/d)^2 + 16*b*c*d*tan(1/2*a)*tan(b*c/d)^2 + 16*d^2*tan(b*
x)*tan(1/2*a)*tan(b*c/d)^2 + 10*d^2*tan(1/2*a)^2*tan(b*c/d)^2 - 4*b^2*c^2*real_part(cos_integral(2*b*x + 2*b*c
/d)) - 4*b^2*c^2*real_part(cos_integral(-2*b*x - 2*b*c/d)) + 8*b*d^2*x*tan(b*x) + 16*b*d^2*x*tan(1/2*a) + 8*b*
c*d*tan(b*x) + d^2*tan(b*x)^2 + 16*b*c*d*tan(1/2*a) + 16*d^2*tan(b*x)*tan(1/2*a) + 10*d^2*tan(1/2*a)^2 - 3*d^2
*tan(b*c/d)^2 - 3*d^2)/(d^5*x^2*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 2*c*d^4*x*tan(b*x)^2*tan(1/2*a)^4*tan(b
*c/d)^2 + d^5*x^2*tan(b*x)^2*tan(1/2*a)^4 + 2*d^5*x^2*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + d^5*x^2*tan(1/2*a
)^4*tan(b*c/d)^2 + c^2*d^3*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 2*c*d^4*x*tan(b*x)^2*tan(1/2*a)^4 + 4*c*d^4*
x*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + 2*c*d^4*x*tan(1/2*a)^4*tan(b*c/d)^2 + 2*d^5*x^2*tan(b*x)^2*tan(1/2*a)
^2 + d^5*x^2*tan(1/2*a)^4 + c^2*d^3*tan(b*x)^2*tan(1/2*a)^4 + d^5*x^2*tan(b*x)^2*tan(b*c/d)^2 + 2*d^5*x^2*tan(
1/2*a)^2*tan(b*c/d)^2 + 2*c^2*d^3*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + c^2*d^3*tan(1/2*a)^4*tan(b*c/d)^2 + 4
*c*d^4*x*tan(b*x)^2*tan(1/2*a)^2 + 2*c*d^4*x*tan(1/2*a)^4 + 2*c*d^4*x*tan(b*x)^2*tan(b*c/d)^2 + 4*c*d^4*x*tan(
1/2*a)^2*tan(b*c/d)^2 + d^5*x^2*tan(b*x)^2 + 2*d^5*x^2*tan(1/2*a)^2 + 2*c^2*d^3*tan(b*x)^2*tan(1/2*a)^2 + c^2*
d^3*tan(1/2*a)^4 + d^5*x^2*tan(b*c/d)^2 + c^2*d^3*tan(b*x)^2*tan(b*c/d)^2 + 2*c^2*d^3*tan(1/2*a)^2*tan(b*c/d)^
2 + 2*c*d^4*x*tan(b*x)^2 + 4*c*d^4*x*tan(1/2*a)^2 + 2*c*d^4*x*tan(b*c/d)^2 + d^5*x^2 + c^2*d^3*tan(b*x)^2 + 2*
c^2*d^3*tan(1/2*a)^2 + c^2*d^3*tan(b*c/d)^2 + 2*c*d^4*x + c^2*d^3)