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

Optimal. Leaf size=129 \[ -\frac{\sin \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b c}{d}+4 b x\right )}{8 d}+\frac{\sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{4 d}+\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{4 d}-\frac{\cos \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{8 d} \]

[Out]

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

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Rubi [A]  time = 0.232436, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4406, 3303, 3299, 3302} \[ -\frac{\sin \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b c}{d}+4 b x\right )}{8 d}+\frac{\sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{4 d}+\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{4 d}-\frac{\cos \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 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 \frac{\cos (a+b x) \sin ^3(a+b x)}{c+d x} \, dx &=\int \left (\frac{\sin (2 a+2 b x)}{4 (c+d x)}-\frac{\sin (4 a+4 b x)}{8 (c+d x)}\right ) \, dx\\ &=-\left (\frac{1}{8} \int \frac{\sin (4 a+4 b x)}{c+d x} \, dx\right )+\frac{1}{4} \int \frac{\sin (2 a+2 b x)}{c+d x} \, dx\\ &=-\left (\frac{1}{8} \cos \left (4 a-\frac{4 b c}{d}\right ) \int \frac{\sin \left (\frac{4 b c}{d}+4 b x\right )}{c+d x} \, dx\right )+\frac{1}{4} \cos \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx-\frac{1}{8} \sin \left (4 a-\frac{4 b c}{d}\right ) \int \frac{\cos \left (\frac{4 b c}{d}+4 b x\right )}{c+d x} \, dx+\frac{1}{4} \sin \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx\\ &=-\frac{\text{Ci}\left (\frac{4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac{4 b c}{d}\right )}{8 d}+\frac{\text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{4 d}+\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{4 d}-\frac{\cos \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b c}{d}+4 b x\right )}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.416103, size = 110, normalized size = 0.85 \[ -\frac{\sin \left (4 a-\frac{4 b c}{d}\right ) \text{CosIntegral}\left (\frac{4 b (c+d x)}{d}\right )-2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )-2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+\cos \left (4 a-\frac{4 b c}{d}\right ) \text{Si}\left (\frac{4 b (c+d x)}{d}\right )}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.021, size = 178, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{b}{8} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 2\,{\frac{-ad+bc}{d}} \right ) }-2\,{\frac{1}{d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 2\,{\frac{-ad+bc}{d}} \right ) } \right ) }-{\frac{b}{32} \left ( 4\,{\frac{1}{d}{\it Si} \left ( 4\,bx+4\,a+4\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 4\,{\frac{-ad+bc}{d}} \right ) }-4\,{\frac{1}{d}{\it Ci} \left ( 4\,bx+4\,a+4\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 4\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/8*b*(2*Si(2*b*x+2*a+2*(-a*d+b*c)/d)*cos(2*(-a*d+b*c)/d)/d-2*Ci(2*b*x+2*a+2*(-a*d+b*c)/d)*sin(2*(-a*d+b*
c)/d)/d)-1/32*b*(4*Si(4*b*x+4*a+4*(-a*d+b*c)/d)*cos(4*(-a*d+b*c)/d)/d-4*Ci(4*b*x+4*a+4*(-a*d+b*c)/d)*sin(4*(-a
*d+b*c)/d)/d))

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Maxima [C]  time = 1.43695, size = 370, normalized size = 2.87 \begin{align*} \frac{b{\left (-2 i \, E_{1}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + 2 i \, E_{1}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + b{\left (i \, E_{1}\left (\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right ) - i \, E_{1}\left (-\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, b{\left (E_{1}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{1}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + b{\left (E_{1}\left (\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right ) + E_{1}\left (-\frac{4 i \, b c + 4 i \,{\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right )}{16 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*(-2*I*exp_integral_e(1, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) + 2*I*exp_integral_e(1, -(2*I*b*c + 2
*I*(b*x + a)*d - 2*I*a*d)/d))*cos(-2*(b*c - a*d)/d) + b*(I*exp_integral_e(1, (4*I*b*c + 4*I*(b*x + a)*d - 4*I*
a*d)/d) - I*exp_integral_e(1, -(4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d))*cos(-4*(b*c - a*d)/d) - 2*b*(exp_inte
gral_e(1, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) + exp_integral_e(1, -(2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/
d))*sin(-2*(b*c - a*d)/d) + b*(exp_integral_e(1, (4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d) + exp_integral_e(1,
-(4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d))*sin(-4*(b*c - a*d)/d))/(b*d)

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Fricas [A]  time = 0.480233, size = 421, normalized size = 3.26 \begin{align*} \frac{2 \,{\left (\operatorname{Ci}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) -{\left (\operatorname{Ci}\left (\frac{4 \,{\left (b d x + b c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{4 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{4 \,{\left (b d x + b c\right )}}{d}\right ) + 4 \, \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right )}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*(cos_integral(2*(b*d*x + b*c)/d) + cos_integral(-2*(b*d*x + b*c)/d))*sin(-2*(b*c - a*d)/d) - (cos_inte
gral(4*(b*d*x + b*c)/d) + cos_integral(-4*(b*d*x + b*c)/d))*sin(-4*(b*c - a*d)/d) - 2*cos(-4*(b*c - a*d)/d)*si
n_integral(4*(b*d*x + b*c)/d) + 4*cos(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*x)**3*cos(a + b*x)/(c + d*x), x)

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Giac [C]  time = 1.72027, size = 8162, normalized size = 63.27 \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)*sin(b*x+a)^3/(d*x+c),x, algorithm="giac")

[Out]

-1/16*(imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*imag_part(
cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(-2*b
*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(2*b*c
/d)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 4*rea
l_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) - 4*real_part(cos_integral
(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(4*b*x + 4*b*c/d))
*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)
^2*tan(2*b*c/d)*tan(b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan
(b*c/d)^2 + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*real
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(
-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(4*b*x + 4*b*c/d))*ta
n(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)
^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 - imag_part(cos_integral(-
4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*t
an(2*b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 - 8*imag_part(cos_integra
l(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) + 8*imag_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)*tan(2*b*c/d
)^2*tan(b*c/d) - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 - 2*imag_part(cos_i
ntegral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2
*a)^2*tan(a)^2*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 - 2*s
in_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2
*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^
2 - 4*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + 8*sin_integral(4
*(b*d*x + b*c)/d)*tan(2*a)*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2
*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b
*c/d)^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_i
ntegral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)
^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - im
ag_part(cos_integral(4*b*x + 4*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(2*b*x +
2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(2*b
*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*sin_
integral(4*(b*d*x + b*c)/d)*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*
tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d) + 2*
real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d) - 4*real_part(cos_integral(2*b*x +
2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2 - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)*tan
(2*b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2 - 2*real_part(cos_in
tegral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2 - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*
a)^2*tan(a)^2*tan(b*c/d) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d) + 4*real
_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x -
2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d) - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(2*b*c
/d)^2*tan(b*c/d) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) + 4*real_par
t(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(b*c/d)^2 + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(2*a)^2*tan(a)*tan(b*c/d)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(b*c/d)^2 + 2
*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(4*b*x + 4
*b*c/d))*tan(2*a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b
*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 - 2*real_pa
rt(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*c
/d))*tan(2*a)*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^
2*tan(b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d)^2 + 4*real_part(c
os_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(4*b*x + 4*b*c/d))*t
an(2*a)^2*tan(a)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2 - 2*imag_part(cos_integral
(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2 - 2*si
n_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2 + 4*
imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d) - 4*imag_part(cos_integral(-4*b*x - 4*
b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d) + 8*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(a)^2*tan(2*b*c/d) + im
ag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*
tan(2*a)^2*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - imag_part(
cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2
*b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2 - imag_part(cos_integral(4*b*x + 4*b*c
/d))*tan(a)^2*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2 - 2*imag_par
t(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2
*tan(2*b*c/d)^2 - 2*sin_integral(4*(b*d*x + b*c)/d)*tan(a)^2*tan(2*b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d
)*tan(a)^2*tan(2*b*c/d)^2 - 8*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(b*c/d) + 8*imag_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)
^2*tan(a)*tan(b*c/d) - 8*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) + 8*imag_pa
rt(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(a
)*tan(2*b*c/d)^2*tan(b*c/d) - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(b*c/d)^2 + 2*imag_part(c
os_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2
*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(b*c/d)^2 - 2*sin_integral(4*(b*d*x +
b*c)/d)*tan(2*a)^2*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(b*c/d)^2 + imag_part(cos_in
tegral(4*b*x + 4*b*c/d))*tan(a)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d
)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - imag_part(cos_integral(-4*b*x - 4*b*
c/d))*tan(a)^2*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*
x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)*tan(b*c/d
)^2 - 4*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)*tan(b*c/d)^2 + 8*sin_integral(4*(b*d*x
+ b*c)/d)*tan(2*a)*tan(2*b*c/d)*tan(b*c/d)^2 - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2*tan(b*
c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(-2*
b*x - 2*b*c/d))*tan(2*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2*tan(b*c
/d)^2 - 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(
2*b*c/d)^2*tan(b*c/d)^2 - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a) - 4*real_part(cos_integ
ral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a) + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2 + 2*re
al_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a
)^2*tan(2*b*c/d) + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - 2*real_part(cos_integ
ral(4*b*x + 4*b*c/d))*tan(a)^2*tan(2*b*c/d) - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2*tan(2*b*c/d
) - 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 2*real_part(cos_integral(-4*b*x - 4*b
*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(2*b*c/d)^2 - 4*real_par
t(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(2*b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^
2*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(b*c/d) - 4*real_part(cos_integral(2*
b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) + 4*real
_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*b*c/d)^2*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*t
an(2*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(b*c/d)^2 + 2*real_part(cos_
integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/
d)^2 + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*
c/d))*tan(2*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)*tan(b*c/d)^2 - imag
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2 - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2 + 2*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 2*
sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2 + imag_part(cos_inte
gral(4*b*x + 4*b*c/d))*tan(a)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 - 2*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(a)^2 - imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2 + 2*sin_integral(4*(b*d*x
+ b*c)/d)*tan(a)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2 + 4*imag_part(cos_integral(4*b*x + 4*b*c/d))*t
an(2*a)*tan(2*b*c/d) - 4*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 8*sin_integral(4*(b
*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d) - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 2*imag_part(c
os_integral(2*b*x + 2*b*c/d))*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*b*c/d)^2 + im
ag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - 4*
sin_integral(2*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - 8*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)
+ 8*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*ta
n(b*c/d) + imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*
tan(b*c/d)^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - imag_part(cos_integral(-4*b*x - 4*b*
c/d))*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(b*
c/d)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*a) - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)
- 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2
*b*c/d) + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(b*c/d) + imag_part(cos_integral(4*b*x + 4*b*c/d)) - 2*imag_part(cos_integral(2*b*x + 2*b*c/d)) + 2*imag_pa
rt(cos_integral(-2*b*x - 2*b*c/d)) - imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 2*sin_integral(4*(b*d*x + b*c
)/d) - 4*sin_integral(2*(b*d*x + b*c)/d))/(d*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(2*a)^2*ta
n(a)^2*tan(2*b*c/d)^2 + d*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 + d*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(
a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(2*a)^2*tan(a)^2 + d*tan(2*a)^2*tan(2*b*c/d)^2 + d*tan(a)^2*tan(2*b*c/
d)^2 + d*tan(2*a)^2*tan(b*c/d)^2 + d*tan(a)^2*tan(b*c/d)^2 + d*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(2*a)^2 + d*
tan(a)^2 + d*tan(2*b*c/d)^2 + d*tan(b*c/d)^2 + d)