3.64 \(\int \sin (a+b x) \text{Si}(c+d x) \, dx\)
Optimal. Leaf size=154 \[ -\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\cos (a+b x) \text{Si}(c+d x)}{b}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]
[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) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cos[a + b*x]*SinInte
gral[c + d*x])/b + (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)
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Rubi [A] time = 0.228481, antiderivative size = 154, 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 =
{6511, 4430, 3303, 3299, 3302} \[ -\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\cos (a+b x) \text{Si}(c+d x)}{b}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[Sin[a + b*x]*SinIntegral[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) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cos[a + b*x]*SinInte
gral[c + d*x])/b + (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)
Rule 6511
Int[Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[(Cos[a + b*x]*SinIntegral[c +
d*x])/b, x] + Dist[d/b, Int[(Cos[a + b*x]*Sin[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 \sin (a+b x) \text{Si}(c+d x) \, dx &=-\frac{\cos (a+b x) \text{Si}(c+d x)}{b}+\frac{d \int \frac{\cos (a+b x) \sin (c+d x)}{c+d x} \, dx}{b}\\ &=-\frac{\cos (a+b x) \text{Si}(c+d 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{\cos (a+b x) \text{Si}(c+d 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{\cos (a+b x) \text{Si}(c+d 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{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac{\cos (a+b x) \text{Si}(c+d 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}\\ \end{align*}
Mathematica [C] time = 1.9715, size = 168, normalized size = 1.09 \[ \frac{i e^{-\frac{i (a d+b c)}{d}} \left (e^{2 i a} \text{ExpIntegralEi}\left (\frac{i (b-d) (c+d x)}{d}\right )-e^{2 i a} \text{ExpIntegralEi}\left (\frac{i (b+d) (c+d x)}{d}\right )+4 i e^{\frac{i (a d+b c)}{d}} \cos (a+b x) \text{Si}(c+d x)-e^{\frac{2 i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b-d) (c+d x)}{d}\right )+e^{\frac{2 i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b+d) (c+d x)}{d}\right )\right )}{4 b} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sin[a + b*x]*SinIntegral[c + d*x],x]
[Out]
((I/4)*(-(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] + (4*I)*E^((I*(b*c + a*d))/d)*Cos[a + b*x]*SinIntegral[c + d*x]))/(b*E^((I*(b*c + a*d))/d))
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Maple [A] time = 0.072, size = 282, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{{\it Si} \left ( dx+c \right ) d}{b}\cos \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(Si(d*x+c)*sin(b*x+a),x)
[Out]
(-Si(d*x+c)/b*d*cos(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 Si}\left (d x + c\right ) \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(Si(d*x+c)*sin(b*x+a),x, algorithm="maxima")
[Out]
integrate(Si(d*x + c)*sin(b*x + a), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left (b x + a\right ) \operatorname{Si}\left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(Si(d*x+c)*sin(b*x+a),x, algorithm="fricas")
[Out]
integral(sin(b*x + a)*sin_integral(d*x + c), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (a + b x \right )} \operatorname{Si}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(Si(d*x+c)*sin(b*x+a),x)
[Out]
Integral(sin(a + b*x)*Si(c + d*x), x)
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Giac [C] time = 2.03416, size = 12880, normalized size = 83.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(Si(d*x+c)*sin(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(c
os_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*t
an(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(c
os_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)*ta
n(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*t
an(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_integr
al((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*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*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)*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)*ta
n(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_i
ntegral(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*t
an(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_integra
l(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(c
os_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)*ta
n(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*(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_integra
l(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(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)^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_integ
ral(-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*t
an(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_integ
ral(-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_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_integra
l((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(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_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_par
t(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)*t
an(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*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*real_part(cos_i
ntegral(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_par
t(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(co
s_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_integral
((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_inte
gral(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_i
ntegral(-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(co
s_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)*t
an(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_i
ntegral((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*ta
n(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(
b*x + a)*sin_integral(d*x + c)/b