∫ cos(a + bx)CosIntegral(c + dx)dx

3.135 \(\int \cos (a+b x) \text{CosIntegral}(c+d x) \, dx\)

Optimal. Leaf size=153 \[ -\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b}-\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b}+\frac{\sin (a+b x) \text{CosIntegral}(c+d x)}{b}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]

[Out]

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

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

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

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Rubi [A] time = 0.216809, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6512, 4430, 3303, 3299, 3302} \[ -\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b}-\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b}+\frac{\sin (a+b x) \text{CosIntegral}(c+d x)}{b}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]

Antiderivative was successfully veriﬁed.

[In]

Int[Cos[a + b*x]*CosIntegral[c + d*x],x]

[Out]

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

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

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

Rule 6512

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(Sin[a + b*x]*CosIntegral[c + d

*x])/b, x] - Dist[d/b, Int[(Sin[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 4430

Int[Cos[(c_.) + (d_.)*(x_)]^(q_.)*((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int[E

xpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Cos[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[

p, 0] && IGtQ[q, 0]

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

Mathematica [C] time = 1.57664, size = 153, normalized size = 1. \[ \frac{4 \sin (a+b x) \text{CosIntegral}(c+d x)+i e^{-\frac{i (a d+b c)}{d}} \left (e^{2 i a} \text{ExpIntegralEi}\left (\frac{i (b-d) (c+d x)}{d}\right )+e^{2 i a} \text{ExpIntegralEi}\left (\frac{i (b+d) (c+d x)}{d}\right )-e^{\frac{2 i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b-d) (c+d x)}{d}\right )-e^{\frac{2 i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b+d) (c+d x)}{d}\right )\right )}{4 b} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cos[a + b*x]*CosIntegral[c + d*x],x]

[Out]

((I*(-(E^(((2*I)*b*c)/d)*ExpIntegralEi[((-I)*(b - d)*(c + d*x))/d]) + E^((2*I)*a)*ExpIntegralEi[(I*(b - d)*(c

+ d*x))/d] - E^(((2*I)*b*c)/d)*ExpIntegralEi[((-I)*(b + d)*(c + d*x))/d] + E^((2*I)*a)*ExpIntegralEi[(I*(b + d

)*(c + d*x))/d]))/E^((I*(b*c + a*d))/d) + 4*CosIntegral[c + d*x]*Sin[a + b*x])/(4*b)

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Maple [A] time = 0.074, size = 282, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ({\frac{{\it Ci} \left ( dx+c \right ) d}{b}\sin \left ({\frac{b \left ( dx+c \right ) }{d}}+{\frac{ad-bc}{d}} \right ) }-{\frac{d}{b} \left ({\frac{d}{2} \left ({\frac{1}{d}{\it Si} \left ({\frac{ \left ( b-d \right ) \left ( dx+c \right ) }{d}}+{\frac{ad-bc}{d}}+{\frac{-ad+bc}{d}} \right ) \cos \left ({\frac{-ad+bc}{d}} \right ) }-{\frac{1}{d}{\it Ci} \left ({\frac{ \left ( b-d \right ) \left ( dx+c \right ) }{d}}+{\frac{ad-bc}{d}}+{\frac{-ad+bc}{d}} \right ) \sin \left ({\frac{-ad+bc}{d}} \right ) } \right ) }+{\frac{d}{2} \left ({\frac{1}{d}{\it Si} \left ({\frac{ \left ( b+d \right ) \left ( dx+c \right ) }{d}}+{\frac{ad-bc}{d}}+{\frac{-ad+bc}{d}} \right ) \cos \left ({\frac{-ad+bc}{d}} \right ) }-{\frac{1}{d}{\it Ci} \left ({\frac{ \left ( b+d \right ) \left ( dx+c \right ) }{d}}+{\frac{ad-bc}{d}}+{\frac{-ad+bc}{d}} \right ) \sin \left ({\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(d*x+c)*cos(b*x+a),x)

[Out]

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

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

b*c)/d+(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci((b+d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d)))/d

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(d*x+c)*cos(b*x+a),x, algorithm="maxima")

[Out]

integrate(Ci(d*x + c)*cos(b*x + a), x)

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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cos \left (b x + a\right ) \operatorname{Ci}\left (d x + c\right ), x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(d*x+c)*cos(b*x+a),x, algorithm="fricas")

[Out]

integral(cos(b*x + a)*cos_integral(d*x + c), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(d*x+c)*cos(b*x+a),x)

[Out]

Integral(cos(a + b*x)*Ci(c + d*x), x)

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Giac [C] time = 1.93965, size = 12879, normalized size = 84.18 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(d*x+c)*cos(b*x+a),x, algorithm="giac")

[Out]

-1/4*(imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c +

c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1

/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(-b*x + d*x + c - b*

c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(

cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1

/2*(b*c - c*d)/d)^2 + 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*

tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a +

1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(b*x

- d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)

+ 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c +

c*d)/d)^2*tan(1/2*(b*c - c*d)/d) + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1

/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(-b*x - d*x - c - b*

c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(

cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*

(b*c - c*d)/d)^2 - 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*t

an(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a

+ 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(cos_integral(-b*

x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^

2 + imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*

d)/d)^2 - imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*

c + c*d)/d)^2 + imag_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(

1/2*(b*c + c*d)/d)^2 - imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)

^2*tan(1/2*(b*c + c*d)/d)^2 + 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1

/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 - 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*

a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + 4*imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*

tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) - 4*imag_part(cos_integral(-b*x + d*x + c -

b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) + 8*sin_integ

ral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b

*c - c*d)/d) - imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/

2*(b*c - c*d)/d)^2 + imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*

tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/

2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2

*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)^2*ta

n(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a + 1/2*c)

^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + 4*imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a

+ 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 4*imag_part(cos_integral(-b*x

- d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 +

8*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*t

an(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c

*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/

2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*

c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1

/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*sin_integral((b*d*x + d^2*x + b*c + c*d)

/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*sin_integral((b*d*x - d^2*x + b

*c - c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(b

*x + d*x + c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_

integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - imag

_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/

d)^2 + imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(

b*c - c*d)/d)^2 - 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*

tan(1/2*(b*c - c*d)/d)^2 + 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c

*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(

1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d) + 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)

^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d) + 2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a +

1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2 + 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1

/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2 - 2*real_part(cos_integral(b*x + d*x + c + b*c/d))

*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 - 2*real_part(cos_integral(-b*x - d*x - c -

b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + 2*real_part(cos_integral(b*x - d*x

- c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d) + 2*real_part(cos_integral(-b*x

+ d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d) - 2*real_part(cos_integr

al(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) - 2*real_part(

cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) + 2

*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c

*d)/d) + 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1

/2*(b*c - c*d)/d) - 2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*t

an(1/2*(b*c - c*d)/d)^2 - 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1

/2*c)*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*a

- 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*ta

n(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2

*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(

1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(cos_integral(b*x + d*x + c + b*

c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(cos_integral(-b*x - d

*x - c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(cos_integr

al(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(

cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2

*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d

)/d)^2 - 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2

*(b*c - c*d)/d)^2 - imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2 -

imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2 + imag_part(cos_inte

gral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2 + imag_part(cos_integral(-b*x - d*x -

c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2 - 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*

a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2 - 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2

*a - 1/2*c)^2 + 4*imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1

/2*(b*c + c*d)/d) - 4*imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*

tan(1/2*(b*c + c*d)/d) + 8*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2

*tan(1/2*(b*c + c*d)/d) + imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c

*d)/d)^2 + imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 - imag

_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 - imag_part(cos_inte

gral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + 2*sin_integral((b*d*x + d^2*x +

b*c + c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*ta

n(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 - imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a - 1/2*c

)^2*tan(1/2*(b*c + c*d)/d)^2 - imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*

c + c*d)/d)^2 + imag_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2

+ imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 - 2*sin_integr

al((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 - 2*sin_integral((b*d*x - d^2*

x + b*c - c*d)/d)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + 4*imag_part(cos_integral(b*x - d*x - c + b*c

/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d) - 4*imag_part(cos_integral(-b*x + d*x + c

- b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d) + 8*sin_integral((b*d*x - d^2*x + b*c

- c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d) + 4*imag_part(cos_integral(b*x - d*x

- c + b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) - 4*imag_part(cos_integral(-

b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) + 8*sin_integral((b

*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a - 1/2*c)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) - imag_part(co

s_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(b*x

- d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(-b*x + d*x + c - b*

c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*

a + 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)^2*tan

(1/2*(b*c - c*d)/d)^2 - 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c - c*d)

/d)^2 + imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + imag_pa

rt(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral

(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(-b*x - d*x -

c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(

1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a - 1/2*c)^2

*tan(1/2*(b*c - c*d)/d)^2 + 4*imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*(b*c +

c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 4*imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan(1

/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 + 8*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)*

tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*(b*c

+ c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*(b*c + c*d)/d)^2

*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(

b*c - c*d)/d)^2 + imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)

/d)^2 - 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*si

n_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_

integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c) + 2*real_part(cos_integral(-b*x + d*x

+ c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c) + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(

1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2 + 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan

(1/2*a - 1/2*c)^2 + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/

d) + 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d) - 2*real_pa

rt(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d) - 2*real_part(cos_integral

(-b*x - d*x - c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d) - 2*real_part(cos_integral(b*x + d*x + c

+ b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*(b*c + c*d)/d)^2 - 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan

(1/2*a + 1/2*c)*tan(1/2*(b*c + c*d)/d)^2 + 2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)

*tan(1/2*(b*c + c*d)/d)^2 + 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c

+ c*d)/d)^2 - 2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c - c*d)/d) - 2

*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c - c*d)/d) + 2*real_part(cos

_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d) + 2*real_part(cos_integral(-b*x

+ d*x + c - b*c/d))*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d) - 2*real_part(cos_integral(b*x - d*x - c + b*c

/d))*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) - 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1

/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d) + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*

c)*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*(b*

c - c*d)/d)^2 - 2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d)^2 -

2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(c

os_integral(b*x + d*x + c + b*c/d))*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 2*real_part(cos_integral

(-b*x - d*x - c - b*c/d))*tan(1/2*(b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(b*x + d*x +

c + b*c/d))*tan(1/2*a + 1/2*c)^2 + imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a + 1/2*c)^2 - imag

_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a + 1/2*c)^2 + imag_part(cos_integral(-b*x - d*x - c - b*c

/d))*tan(1/2*a + 1/2*c)^2 - 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)^2 + 2*sin_integra

l((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*a + 1/2*c)^2 + imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2

*a - 1/2*c)^2 - imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)^2 + imag_part(cos_integral(-

b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c)^2 - imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a - 1/2

*c)^2 + 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a - 1/2*c)^2 - 2*sin_integral((b*d*x - d^2*x + b

*c - c*d)/d)*tan(1/2*a - 1/2*c)^2 + 4*imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1/2*c)*tan(1/

2*(b*c + c*d)/d) - 4*imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c)*tan(1/2*(b*c + c*d)/d)

+ 8*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*a + 1/2*c)*tan(1/2*(b*c + c*d)/d) - imag_part(cos_int

egral(b*x + d*x + c + b*c/d))*tan(1/2*(b*c + c*d)/d)^2 + imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/

2*(b*c + c*d)/d)^2 - imag_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*(b*c + c*d)/d)^2 + imag_part(cos_

integral(-b*x - d*x - c - b*c/d))*tan(1/2*(b*c + c*d)/d)^2 - 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan

(1/2*(b*c + c*d)/d)^2 + 2*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*tan(1/2*(b*c + c*d)/d)^2 + 4*imag_part(c

os_integral(b*x - d*x - c + b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d) - 4*imag_part(cos_integral(-b*x

+ d*x + c - b*c/d))*tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d) + 8*sin_integral((b*d*x - d^2*x + b*c - c*d)/d)*

tan(1/2*a - 1/2*c)*tan(1/2*(b*c - c*d)/d) + imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*(b*c - c*d)

/d)^2 - imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*(b*c - c*d)/d)^2 + imag_part(cos_integral(-b*x

+ d*x + c - b*c/d))*tan(1/2*(b*c - c*d)/d)^2 - imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*(b*c -

c*d)/d)^2 + 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 - 2*sin_integral((b*d*x - d

^2*x + b*c - c*d)/d)*tan(1/2*(b*c - c*d)/d)^2 + 2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*a + 1

/2*c) + 2*real_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*a + 1/2*c) + 2*real_part(cos_integral(b*x -

d*x - c + b*c/d))*tan(1/2*a - 1/2*c) + 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*a - 1/2*c) -

2*real_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*(b*c + c*d)/d) - 2*real_part(cos_integral(-b*x - d*x

- c - b*c/d))*tan(1/2*(b*c + c*d)/d) - 2*real_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*(b*c - c*d)/d)

- 2*real_part(cos_integral(-b*x + d*x + c - b*c/d))*tan(1/2*(b*c - c*d)/d) + imag_part(cos_integral(b*x + d*x

+ c + b*c/d)) + imag_part(cos_integral(b*x - d*x - c + b*c/d)) - imag_part(cos_integral(-b*x + d*x + c - b*c/

d)) - imag_part(cos_integral(-b*x - d*x - c - b*c/d)) + 2*sin_integral((b*d*x + d^2*x + b*c + c*d)/d) + 2*sin_

integral((b*d*x - d^2*x + b*c - c*d)/d))*d/(b*d*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/

d)^2*tan(1/2*(b*c - c*d)/d)^2 + b*d*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + b*d*t

an(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + b*d*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*

d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + b*d*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2

+ b*d*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2 + b*d*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + b*d*tan(

1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2 + b*d*tan(1/2*a + 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + b*d*tan(1/2*a

- 1/2*c)^2*tan(1/2*(b*c - c*d)/d)^2 + b*d*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + b*d*tan(1/2*a +

1/2*c)^2 + b*d*tan(1/2*a - 1/2*c)^2 + b*d*tan(1/2*(b*c + c*d)/d)^2 + b*d*tan(1/2*(b*c - c*d)/d)^2 + b*d) + cos

_integral(d*x + c)*sin(b*x + a)/b

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