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

Optimal. Leaf size=149 \[ -\frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^2}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^2}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^2}+\frac{\cos (c) \text{Si}(d x)}{a^2}-\frac{d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a b}+\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a b}+\frac{\sin (c+d x)}{a (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.410201, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3303, 3299, 3302, 3297} \[ -\frac{\sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a^2}-\frac{\cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a^2}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^2}+\frac{\cos (c) \text{Si}(d x)}{a^2}-\frac{d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{a b}+\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{a b}+\frac{\sin (c+d x)}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x)^2),x]

[Out]

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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]

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]

Rubi steps

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

Mathematica [C]  time = 4.20623, size = 641, normalized size = 4.3 \[ \frac{e^{-\frac{i d (2 a+b x)}{b}} \left (i a^2 d \sin (c) e^{\frac{i d (3 a+b x)}{b}} \text{Ei}\left (-\frac{i d (a+b x)}{b}\right )-i a^2 d \sin (c) e^{\frac{i d (a+b x)}{b}} \text{Ei}\left (\frac{i d (a+b x)}{b}\right )+a^2 (-d) \cos (c) e^{\frac{i d (3 a+b x)}{b}} \text{Ei}\left (-\frac{i d (a+b x)}{b}\right )-a^2 d \cos (c) e^{\frac{i d (a+b x)}{b}} \text{Ei}\left (\frac{i d (a+b x)}{b}\right )+2 b^2 x \cos (c) \text{Si}(d x) e^{\frac{i d (2 a+b x)}{b}}-2 b^2 x e^{\frac{i d (2 a+b x)}{b}} \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )+2 b \sin (c) (a+b x) \text{CosIntegral}(d x) e^{\frac{i d (2 a+b x)}{b}}-2 b (a+b x) e^{\frac{i d (2 a+b x)}{b}} \sin \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )+i a b d x \sin (c) e^{\frac{i d (3 a+b x)}{b}} \text{Ei}\left (-\frac{i d (a+b x)}{b}\right )-i a b d x \sin (c) e^{\frac{i d (a+b x)}{b}} \text{Ei}\left (\frac{i d (a+b x)}{b}\right )-a b d x \cos (c) e^{\frac{i d (3 a+b x)}{b}} \text{Ei}\left (-\frac{i d (a+b x)}{b}\right )-a b d x \cos (c) e^{\frac{i d (a+b x)}{b}} \text{Ei}\left (\frac{i d (a+b x)}{b}\right )+2 a b \cos (c) \text{Si}(d x) e^{\frac{i d (2 a+b x)}{b}}-2 a b e^{\frac{i d (2 a+b x)}{b}} \cos \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )+a b \sin (c) e^{\frac{2 i d (a+b x)}{b}}-i a b \cos (c) e^{\frac{2 i d (a+b x)}{b}}+a b \sin (c) e^{\frac{2 i a d}{b}}+i a b \cos (c) e^{\frac{2 i a d}{b}}\right )}{2 a^2 b (a+b x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(I*a*b*E^(((2*I)*a*d)/b)*Cos[c] - I*a*b*E^(((2*I)*d*(a + b*x))/b)*Cos[c] - a^2*d*E^((I*d*(3*a + b*x))/b)*Cos[c
]*ExpIntegralEi[((-I)*d*(a + b*x))/b] - a*b*d*E^((I*d*(3*a + b*x))/b)*x*Cos[c]*ExpIntegralEi[((-I)*d*(a + b*x)
)/b] - a^2*d*E^((I*d*(a + b*x))/b)*Cos[c]*ExpIntegralEi[(I*d*(a + b*x))/b] - a*b*d*E^((I*d*(a + b*x))/b)*x*Cos
[c]*ExpIntegralEi[(I*d*(a + b*x))/b] + a*b*E^(((2*I)*a*d)/b)*Sin[c] + a*b*E^(((2*I)*d*(a + b*x))/b)*Sin[c] + 2
*b*E^((I*d*(2*a + b*x))/b)*(a + b*x)*CosIntegral[d*x]*Sin[c] + I*a^2*d*E^((I*d*(3*a + b*x))/b)*ExpIntegralEi[(
(-I)*d*(a + b*x))/b]*Sin[c] + I*a*b*d*E^((I*d*(3*a + b*x))/b)*x*ExpIntegralEi[((-I)*d*(a + b*x))/b]*Sin[c] - I
*a^2*d*E^((I*d*(a + b*x))/b)*ExpIntegralEi[(I*d*(a + b*x))/b]*Sin[c] - I*a*b*d*E^((I*d*(a + b*x))/b)*x*ExpInte
gralEi[(I*d*(a + b*x))/b]*Sin[c] - 2*b*E^((I*d*(2*a + b*x))/b)*(a + b*x)*CosIntegral[d*(a/b + x)]*Sin[c - (a*d
)/b] + 2*a*b*E^((I*d*(2*a + b*x))/b)*Cos[c]*SinIntegral[d*x] + 2*b^2*E^((I*d*(2*a + b*x))/b)*x*Cos[c]*SinInteg
ral[d*x] - 2*a*b*E^((I*d*(2*a + b*x))/b)*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 2*b^2*E^((I*d*(2*a + b*x)
)/b)*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(2*a^2*b*E^((I*d*(2*a + b*x))/b)*(a + b*x))

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Maple [A]  time = 0.014, size = 210, normalized size = 1.4 \begin{align*} -{\frac{bd}{a} \left ( -{\frac{\sin \left ( dx+c \right ) }{ \left ( \left ( dx+c \right ) b+da-cb \right ) b}}+{\frac{1}{b} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) }+{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) } \right ) } \right ) }-{\frac{b}{{a}^{2}} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) }-{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) } \right ) }+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.82896, size = 682, normalized size = 4.58 \begin{align*} \frac{2 \, a b \sin \left (d x + c\right ) + 2 \,{\left (b^{2} x + a b\right )} \cos \left (c\right ) \operatorname{Si}\left (d x\right ) -{\left ({\left (a b d x + a^{2} d\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (a b d x + a^{2} d\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) + 2 \,{\left (b^{2} x + a b\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) +{\left ({\left (b^{2} x + a b\right )} \operatorname{Ci}\left (d x\right ) +{\left (b^{2} x + a b\right )} \operatorname{Ci}\left (-d x\right )\right )} \sin \left (c\right ) +{\left ({\left (b^{2} x + a b\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (b^{2} x + a b\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right ) - 2 \,{\left (a b d x + a^{2} d\right )} \operatorname{Si}\left (\frac{b d x + a d}{b}\right )\right )} \sin \left (-\frac{b c - a d}{b}\right )}{2 \,{\left (a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*b*sin(d*x + c) + 2*(b^2*x + a*b)*cos(c)*sin_integral(d*x) - ((a*b*d*x + a^2*d)*cos_integral((b*d*x +
a*d)/b) + (a*b*d*x + a^2*d)*cos_integral(-(b*d*x + a*d)/b) + 2*(b^2*x + a*b)*sin_integral((b*d*x + a*d)/b))*co
s(-(b*c - a*d)/b) + ((b^2*x + a*b)*cos_integral(d*x) + (b^2*x + a*b)*cos_integral(-d*x))*sin(c) + ((b^2*x + a*
b)*cos_integral((b*d*x + a*d)/b) + (b^2*x + a*b)*cos_integral(-(b*d*x + a*d)/b) - 2*(a*b*d*x + a^2*d)*sin_inte
gral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(a^2*b^2*x + a^3*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.46255, size = 10037, normalized size = 67.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b*d*x*real
_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*d*x*imag_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))
*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c
)^2*tan(1/2*a*d/b) + 2*a*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2
 - 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a*b*d*x*sin_
integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + b^2*x*imag_part(cos_integral(d*x + a*d/
b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 - b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^
2 - b^2*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*d*real_part(cos_int
egral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*d*real_part(cos_integral(-d*x - a*d/b))
*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*x*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*
a*d/b)^2 + 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*b*d*x*real_p
art(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan
(1/2*d*x)^2*tan(1/2*c)^2 + 4*a*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*
d/b) + 4*a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 2*a^2*d*imag
_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d*imag_part(cos_integral(-
d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/
2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)
^2*tan(1/2*a*d/b) - 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - a*b*d*x
*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a*b*d*x*real_part(cos_integral(-d*x -
a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^2*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*
c)*tan(1/2*a*d/b)^2 - 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2
 - 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*x*real_part
(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*x*real_part(cos_integral(-d*x - a*d/b))
*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)*
tan(1/2*a*d/b)^2 + 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + a*b*d*x*
real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b*d*x*real_part(cos_integral(-d*x - a*d
/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(
1/2*a*d/b)^2 + a*b*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*b*imag_part(c
os_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*b*imag_part(cos_integral(-d*x))*ta
n(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b
)^2 + 2*a*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*d*x*imag_part(c
os_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*
d*x)^2*tan(1/2*c) - 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) - b^2*x*imag_part(cos_in
tegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^2*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c
)^2 + b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - b^2*x*imag_part(cos_integral(-
d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a
^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*b^2*x*sin_integral(d*x)*tan(1/2*d*x
)^2*tan(1/2*c)^2 - 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b*d*x*imag_part(cos
_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/
2*d*x)^2*tan(1/2*a*d/b) + 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*b^2*x*imag
_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*b^2*x*imag_part(cos_integral(-d*
x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*
x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1
/2*a*d/b) + 8*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 2*a*b*d*x*imag_pa
rt(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*c)^2*tan(1/2*a*d/b) - 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a*b*real_p
art(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a*b*real_part(cos_integral(-d*x
- a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)
^2*tan(1/2*a*d/b)^2 - b^2*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^2*x*imag_part(cos
_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^2*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*
tan(1/2*a*d/b)^2 - a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a^2*d*real_par
t(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*b^2*x*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1
/2*a*d/b)^2 - 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a*b*d*x*imag_part(cos_
integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c
)*tan(1/2*a*d/b)^2 + 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*real_part(cos
_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*real_part(cos_integral(d*x))*tan(1/
2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*t
an(1/2*a*d/b)^2 - 2*a*b*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + b^2*x*imag_
part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*x*imag_part(cos_integral(d*x))*tan(1/2*c)^
2*tan(1/2*a*d/b)^2 - b^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*x*imag_pa
rt(cos_integral(-d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2
*tan(1/2*a*d/b)^2 + a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*x*sin_in
tegral(d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^
2 + a*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a*b*d*x*real_part(cos_integral(-d*x - a*d/b)
)*tan(1/2*d*x)^2 - 2*a^2*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d*imag_part(
cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d
*x)^2*tan(1/2*c) - 2*b^2*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 2*b^2*x*real_part(cos_inte
gral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*b^2*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c
) - 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) - a*b*d*x*real_part(cos_integral(d*x + a*d
/b))*tan(1/2*c)^2 - a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - a*b*imag_part(cos_integral(d*
x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*b*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*b*i
mag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*b*imag_part(cos_integral(-d*x))*tan(1/2*d
*x)^2*tan(1/2*c)^2 + 2*a*b*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a*b*sin_integral((b*d*x + a*d)/b)
*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*
a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*b^2*x*real_part(cos_integral(d*x
 + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/
2*a*d/b) + 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a*b*d*x*real_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1
/2*a*d/b) + 4*a*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*a*b*imag_p
art(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*a*b*sin_integral((b*d*x + a*d)/b)
*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 2*a^2*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*
a*d/b) + 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*x*real_part(cos_int
egral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*t
an(1/2*a*d/b) - 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - a*b*d*x*real_part(cos_inte
gral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - a*b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - a*b*ima
g_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - a*b*imag_part(cos_integral(d*x))*tan(1/2*d
*x)^2*tan(1/2*a*d/b)^2 + a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*b*imag_
part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a*b*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*a*d/
b)^2 - 2*a*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a^2*d*imag_part(cos_integral(d*
x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d
/b)^2 - 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*x*real_part(cos_integ
ral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b
)^2 - 2*b^2*x*real_part(cos_integral(-d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a^2*d*sin_integral((b*d*x + a*d)/b
)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a*b*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + a*b*imag_part(cos_integral(
d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b*imag_part(cos_integral(d*x))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 -
 a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*b*imag_part(cos_integral(-d*x))*t
an(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*sin_integral(d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*sin_integral((b*d
*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a*b*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*x*imag_par
t(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 - b^2*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 - b^2*x*imag_p
art(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + b^2*x*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 + a^2*d*re
al_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2
 - 2*b^2*x*sin_integral(d*x)*tan(1/2*d*x)^2 + 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 - 2*a*b*d*x
*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)
- 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c) + 2*a*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)
^2*tan(1/2*c) - 2*a*b*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*b*real_part(cos_integral(-d
*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*b*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - b^2*x
*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 + b^2*x*imag_part(cos_integral(d*x))*tan(1/2*c)^2 + b^2*x*i
mag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - b^2*x*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 - a^2*d*r
eal_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 +
2*b^2*x*sin_integral(d*x)*tan(1/2*c)^2 - 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*a*b*d*x*imag_p
art(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)
 + 4*a*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) - 2*a*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2
*d*x)^2*tan(1/2*a*d/b) - 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*b^2*x*i
mag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*b^2*x*imag_part(cos_integral(-d*x - a*d/b))*
tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*d*r
eal_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2
*c)*tan(1/2*a*d/b) + 2*a*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a*b*real_part(
cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - b^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a
*d/b)^2 - b^2*x*imag_part(cos_integral(d*x))*tan(1/2*a*d/b)^2 + b^2*x*imag_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*a*d/b)^2 + b^2*x*imag_part(cos_integral(-d*x))*tan(1/2*a*d/b)^2 - a^2*d*real_part(cos_integral(d*x + a*d
/b))*tan(1/2*a*d/b)^2 - a^2*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 2*b^2*x*sin_integral(d*
x)*tan(1/2*a*d/b)^2 - 2*b^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 2*a*b*real_part(cos_integral(d*
x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*real_part(cos_integral(d*x))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a
*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*real_part(cos_integral(-d*x))*tan
(1/2*c)*tan(1/2*a*d/b)^2 + a*b*d*x*real_part(cos_integral(d*x + a*d/b)) + a*b*d*x*real_part(cos_integral(-d*x
- a*d/b)) + a*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 - a*b*imag_part(cos_integral(d*x))*tan(1/2
*d*x)^2 - a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + a*b*imag_part(cos_integral(-d*x))*tan(1/2
*d*x)^2 - 2*a*b*sin_integral(d*x)*tan(1/2*d*x)^2 + 2*a*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 - 2*a^2*
d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) +
 2*b^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) - 2*b^2*x*real_part(cos_integral(d*x))*tan(1/2*c) + 2
*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*b^2*x*real_part(cos_integral(-d*x))*tan(1/2*c) - 4
*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*c) + 4*a*b*tan(1/2*d*x)^2*tan(1/2*c) - a*b*imag_part(cos_integral
(d*x + a*d/b))*tan(1/2*c)^2 + a*b*imag_part(cos_integral(d*x))*tan(1/2*c)^2 + a*b*imag_part(cos_integral(-d*x
- a*d/b))*tan(1/2*c)^2 - a*b*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a*b*sin_integral(d*x)*tan(1/2*c)^2
 - 2*a*b*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 4*a*b*tan(1/2*d*x)*tan(1/2*c)^2 + 2*a^2*d*imag_part(cos_
integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a^2*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) - 2*b^2*x
*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*b^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a
*d/b) + 4*a^2*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) + 4*a*b*imag_part(cos_integral(d*x + a*d/b))*tan(
1/2*c)*tan(1/2*a*d/b) - 4*a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*a*b*sin_inte
gral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - a*b*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 -
a*b*imag_part(cos_integral(d*x))*tan(1/2*a*d/b)^2 + a*b*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2
 + a*b*imag_part(cos_integral(-d*x))*tan(1/2*a*d/b)^2 - 2*a*b*sin_integral(d*x)*tan(1/2*a*d/b)^2 - 2*a*b*sin_i
ntegral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 4*a*b*tan(1/2*d*x)*tan(1/2*a*d/b)^2 - 4*a*b*tan(1/2*c)*tan(1/2*a*d
/b)^2 + b^2*x*imag_part(cos_integral(d*x + a*d/b)) - b^2*x*imag_part(cos_integral(d*x)) - b^2*x*imag_part(cos_
integral(-d*x - a*d/b)) + b^2*x*imag_part(cos_integral(-d*x)) + a^2*d*real_part(cos_integral(d*x + a*d/b)) + a
^2*d*real_part(cos_integral(-d*x - a*d/b)) - 2*b^2*x*sin_integral(d*x) + 2*b^2*x*sin_integral((b*d*x + a*d)/b)
 + 2*a*b*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) - 2*a*b*real_part(cos_integral(d*x))*tan(1/2*c) + 2*a
*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 2*a*b*real_part(cos_integral(-d*x))*tan(1/2*c) - 2*a*b*r
eal_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a*b*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b
) + a*b*imag_part(cos_integral(d*x + a*d/b)) - a*b*imag_part(cos_integral(d*x)) - a*b*imag_part(cos_integral(-
d*x - a*d/b)) + a*b*imag_part(cos_integral(-d*x)) - 2*a*b*sin_integral(d*x) + 2*a*b*sin_integral((b*d*x + a*d)
/b) - 4*a*b*tan(1/2*d*x) - 4*a*b*tan(1/2*c))*b/(a^2*b^3*x*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*b
^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*b^3*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^2*b^3*x*tan(1/2*d*
x)^2*tan(1/2*a*d/b)^2 + a^2*b^3*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^3*b^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^3*b^
2*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a^3*b^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*b^3*x*tan(1/2*d*x)^2 + a^2*b^3
*x*tan(1/2*c)^2 + a^2*b^3*x*tan(1/2*a*d/b)^2 + a^3*b^2*tan(1/2*d*x)^2 + a^3*b^2*tan(1/2*c)^2 + a^3*b^2*tan(1/2
*a*d/b)^2 + a^2*b^3*x + a^3*b^2)

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