∫ cos(a + bx)Si(c + dx)dx

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

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

[Out]

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

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

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

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Rubi [A] time = 0.222689, 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 = {6517, 4428, 3303, 3299, 3302} \[ -\frac{\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}+\frac{\sin (a+b x) \text{Si}(c+d x)}{b}-\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

Rule 6517

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

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

Rule 4428

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

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

0] && IGtQ[q, 0] && IntegerQ[m]

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{Si}(c+d x) \, dx &=\frac{\sin (a+b x) \text{Si}(c+d x)}{b}-\frac{d \int \frac{\sin (a+b x) \sin (c+d x)}{c+d x} \, dx}{b}\\ &=\frac{\sin (a+b x) \text{Si}(c+d x)}{b}-\frac{d \int \left (\frac{\cos (a-c+(b-d) x)}{2 (c+d x)}-\frac{\cos (a+c+(b+d) x)}{2 (c+d x)}\right ) \, dx}{b}\\ &=\frac{\sin (a+b x) \text{Si}(c+d x)}{b}-\frac{d \int \frac{\cos (a-c+(b-d) x)}{c+d x} \, dx}{2 b}+\frac{d \int \frac{\cos (a+c+(b+d) x)}{c+d x} \, dx}{2 b}\\ &=\frac{\sin (a+b x) \text{Si}(c+d x)}{b}-\frac{\left (d \cos \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 \cos \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{\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{\sin \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}\\ &=-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Ci}\left (\frac{c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Ci}\left (\frac{c (b+d)}{d}+(b+d) x\right )}{2 b}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac{\sin (a+b x) \text{Si}(c+d x)}{b}-\frac{\sin \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 = 1.74471, size = 164, normalized size = 1.07 \[ \frac{e^{-\frac{i (a d+b c)}{d}} \left (-e^{2 i a} \text{ExpIntegralEi}\left (\frac{i (b-d) (c+d x)}{d}\right )+e^{2 i a} \text{ExpIntegralEi}\left (\frac{i (b+d) (c+d x)}{d}\right )+4 e^{\frac{i (a d+b c)}{d}} \sin (a+b x) \text{Si}(c+d x)-e^{\frac{2 i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b-d) (c+d x)}{d}\right )+e^{\frac{2 i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b+d) (c+d x)}{d}\right )\right )}{4 b} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A] time = 0.079, size = 280, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ({\frac{{\it Si} \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 ) \sin \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 ) \cos \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 ) \sin \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 ) \cos \left ({\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Si(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)*sin((-a

*d+b*c)/d)/d+Ci((b-d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*cos((-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)*sin((-a*d+b*c)/d)/d+Ci((b+d)/d*(d*x+c)+(a*d-b*c)/d+(-a*d+b*c)/d)*cos((-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 ) \cos \left (b x + a\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Si(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{Si}\left (d x + c\right ), x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(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 \cos{\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(cos(b*x+a)*Si(d*x+c),x)

[Out]

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

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Giac [C] time = 1.97024, size = 12439, normalized size = 81.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - 4*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)^2*tan(1/2*(b*c - c*d)/d)^2 + 2*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)^2*tan(1/2*(b*c - c*d)/d)^2 - 2*ima

g_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 + 4*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)^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))*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 + 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 - 4*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) -

4*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) - 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)^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 + 4*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)*tan(1/2*(b*c - c*d)/d)^2 + 4*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)*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*ta

n(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*(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 - 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*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) - 4*sin_in

tegral((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*ima

g_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*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 + 4*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)^2 + 2*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)^2 - 2*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)^2 + 4*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)^2 + 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) + 4*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*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) - 4*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) + 4*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*a - 1/2*c)*tan(1/2*(b*c - c*d)/d)^2 + 2*ima

g_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

- 4*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)

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

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

an(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*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)*tan(1/2*(b*c - c*d)/d)^2 + 2*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)^2 - 2*imag_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 + 4*sin_integr

al((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)^2 - 2*ima

g_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*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)^2 - 4*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)^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_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 + 4*real_part(cos_integral(b*x + d*x + c + b*c/d))*ta

n(1/2*a + 1/2*c)*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d) + 4*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) + 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*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_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 - 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 - 4*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) - 4*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) - 4*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) - 4*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) - 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*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 + 4*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)*tan(1/2*(b*c - c*d)/d)^2 + 4*real_part(cos_inte

gral(-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 - real_part(c

os_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*(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 + 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*t

an(1/2*a - 1/2*c) + 4*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*

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 + 2*imag_part(cos_integ

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

- 2*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 + 4*sin_integ

ral((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 - 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) - 4*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) + 4*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(c

os_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) - 4*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*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 + 2*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 - 4*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 - 2*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 +

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

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

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

s_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 - 4*real_p

art(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*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) + 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 + rea

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

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

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

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

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

real_part(cos_integral(-b*x + d*x + c - b*c/d)) + real_part(cos_integral(-b*x - d*x - c - b*c/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*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*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) + sin(b*x + a)*sin_integral(d*x + c)/b

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