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

Optimal. Leaf size=129 \[ -\frac{\sin \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b c}{d}+6 b x\right )}{32 d}+\frac{3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{32 d}+\frac{3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{32 d}-\frac{\cos \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b c}{d}+6 b x\right )}{32 d} \]

[Out]

-(CosIntegral[(6*b*c)/d + 6*b*x]*Sin[6*a - (6*b*c)/d])/(32*d) + (3*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2
*b*c)/d])/(32*d) + (3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(32*d) - (Cos[6*a - (6*b*c)/d]*SinI
ntegral[(6*b*c)/d + 6*b*x])/(32*d)

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Rubi [A]  time = 0.246071, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4406, 3303, 3299, 3302} \[ -\frac{\sin \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b c}{d}+6 b x\right )}{32 d}+\frac{3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{32 d}+\frac{3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{32 d}-\frac{\cos \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b c}{d}+6 b x\right )}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(CosIntegral[(6*b*c)/d + 6*b*x]*Sin[6*a - (6*b*c)/d])/(32*d) + (3*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2
*b*c)/d])/(32*d) + (3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(32*d) - (Cos[6*a - (6*b*c)/d]*SinI
ntegral[(6*b*c)/d + 6*b*x])/(32*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 ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx &=\int \left (\frac{3 \sin (2 a+2 b x)}{32 (c+d x)}-\frac{\sin (6 a+6 b x)}{32 (c+d x)}\right ) \, dx\\ &=-\left (\frac{1}{32} \int \frac{\sin (6 a+6 b x)}{c+d x} \, dx\right )+\frac{3}{32} \int \frac{\sin (2 a+2 b x)}{c+d x} \, dx\\ &=-\left (\frac{1}{32} \cos \left (6 a-\frac{6 b c}{d}\right ) \int \frac{\sin \left (\frac{6 b c}{d}+6 b x\right )}{c+d x} \, dx\right )+\frac{1}{32} \left (3 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx-\frac{1}{32} \sin \left (6 a-\frac{6 b c}{d}\right ) \int \frac{\cos \left (\frac{6 b c}{d}+6 b x\right )}{c+d x} \, dx+\frac{1}{32} \left (3 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx\\ &=-\frac{\text{Ci}\left (\frac{6 b c}{d}+6 b x\right ) \sin \left (6 a-\frac{6 b c}{d}\right )}{32 d}+\frac{3 \text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{32 d}+\frac{3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{32 d}-\frac{\cos \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b c}{d}+6 b x\right )}{32 d}\\ \end{align*}

Mathematica [A]  time = 0.309157, size = 110, normalized size = 0.85 \[ -\frac{\sin \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b (c+d x)}{d}\right )-3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )-3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+\cos \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b (c+d x)}{d}\right )}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(CosIntegral[(6*b*(c + d*x))/d]*Sin[6*a - (6*b*c)/d] - 3*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d]
- 3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + Cos[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d])
/(32*d)

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Maple [A]  time = 0.024, size = 178, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{b}{192} \left ( 6\,{\frac{1}{d}{\it Si} \left ( 6\,bx+6\,a+6\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 6\,{\frac{-ad+bc}{d}} \right ) }-6\,{\frac{1}{d}{\it Ci} \left ( 6\,bx+6\,a+6\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 6\,{\frac{-ad+bc}{d}} \right ) } \right ) }+{\frac{3\,b}{64} \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 ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/192*b*(6*Si(6*b*x+6*a+6*(-a*d+b*c)/d)*cos(6*(-a*d+b*c)/d)/d-6*Ci(6*b*x+6*a+6*(-a*d+b*c)/d)*sin(6*(-a*d
+b*c)/d)/d)+3/64*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))

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Maxima [C]  time = 1.5355, size = 370, normalized size = 2.87 \begin{align*} \frac{b{\left (-3 i \, E_{1}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + 3 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{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right ) - i \, E_{1}\left (-\frac{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right )\right )} \cos \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) - 3 \, 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{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right ) + E_{1}\left (-\frac{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right )\right )} \sin \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right )}{64 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(b*(-3*I*exp_integral_e(1, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) + 3*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, (6*I*b*c + 6*I*(b*x + a)*d - 6*I*
a*d)/d) - I*exp_integral_e(1, -(6*I*b*c + 6*I*(b*x + a)*d - 6*I*a*d)/d))*cos(-6*(b*c - a*d)/d) - 3*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, (6*I*b*c + 6*I*(b*x + a)*d - 6*I*a*d)/d) + exp_integral_e(1,
-(6*I*b*c + 6*I*(b*x + a)*d - 6*I*a*d)/d))*sin(-6*(b*c - a*d)/d))/(b*d)

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Fricas [A]  time = 0.490778, size = 421, normalized size = 3.26 \begin{align*} \frac{3 \,{\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{6 \,{\left (b d x + b c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{6 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, \cos \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{6 \,{\left (b d x + b c\right )}}{d}\right ) + 6 \, \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 )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(3*(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(6*(b*d*x + b*c)/d) + cos_integral(-6*(b*d*x + b*c)/d))*sin(-6*(b*c - a*d)/d) - 2*cos(-6*(b*c - a*d)/d)*si
n_integral(6*(b*d*x + b*c)/d) + 6*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 ^{3}{\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)**3*sin(b*x+a)**3/(d*x+c),x)

[Out]

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

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Giac [C]  time = 1.78309, 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)^3*sin(b*x+a)^3/(d*x+c),x, algorithm="giac")

[Out]

-1/64*(imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*imag_part(
cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(-2*b
*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan
(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(3*b*c
/d)^2*tan(b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 6*rea
l_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) - 6*real_part(cos_integral
(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(6*b*x + 6*b*c/d))
*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)
^2*tan(3*b*c/d)*tan(b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan
(b*c/d)^2 + 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*real
_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(
-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(6*b*x + 6*b*c/d))*ta
n(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)
^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 - imag_part(cos_integral(-
6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*t
an(3*b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 - 12*imag_part(cos_integr
al(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) + 12*imag_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)*tan(3*b*c
/d)^2*tan(b*c/d) - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 - 3*imag_part(cos
_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan
(3*a)^2*tan(a)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 - 2
*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)
^2*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)*tan(b*c/d
)^2 - 4*imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + 8*sin_integral
(6*(b*d*x + b*c)/d)*tan(3*a)*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + imag_part(cos_integral(6*b*x + 6*b*c/d))*tan
(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan
(b*c/d)^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos
_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*
a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 -
imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*imag_part(cos_integral(2*b*x
 + 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(3
*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*si
n_integral(6*(b*d*x + b*c)/d)*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^
2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d) +
2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d) - 6*real_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2 - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)*t
an(3*b*c/d)^2 - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2 - 2*real_part(cos_
integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2 - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(
3*a)^2*tan(a)^2*tan(b*c/d) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d) + 6*re
al_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x
- 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d) - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(3*b
*c/d)^2*tan(b*c/d) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) + 6*real_p
art(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(b*c/d)^2 + 6*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(3*a)^2*tan(a)*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(b*c/d)^2 +
 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x +
 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3
*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 - 2*real_
part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(6*b*x + 6*b
*c/d))*tan(3*a)*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(3*b*c/d
)^2*tan(b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*real_part
(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(6*b*x + 6*b*c/d))
*tan(3*a)^2*tan(a)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2 - 3*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2 - 2*
sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2 +
4*imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d) - 4*imag_part(cos_integral(-6*b*x -
6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d) + 8*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)*tan(a)^2*tan(3*b*c/d) +
imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2 - 3*imag_part(cos_integral(2*b*x + 2*b*c/d)
)*tan(3*a)^2*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2 - imag_par
t(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan
(3*b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(3*b*c/d)^2 - imag_part(cos_integral(6*b*x + 6*b
*c/d))*tan(a)^2*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2 - 3*imag_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)
^2*tan(3*b*c/d)^2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(a)^2*tan(3*b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)
/d)*tan(a)^2*tan(3*b*c/d)^2 - 12*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(b*c/d) + 12*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)*tan(
3*a)^2*tan(a)*tan(b*c/d) - 12*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) + 12*i
mag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)
*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(b*c/d)^2 + 3*imag_
part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(b*c/d)^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(
3*a)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(b*c/d)^2 - 2*sin_integral(6*(b*
d*x + b*c)/d)*tan(3*a)^2*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(b*c/d)^2 + imag_part(
cos_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(b*c/d)^2 - 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan
(b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - imag_part(cos_integral(-6*b*x
- 6*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 - 6*sin_integral(2
*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(3*b*c/d)*tan
(b*c/d)^2 - 4*imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(3*b*c/d)*tan(b*c/d)^2 + 8*sin_integral(6*
(b*d*x + b*c)/d)*tan(3*a)*tan(3*b*c/d)*tan(b*c/d)^2 - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*b*c/d)^2*
tan(b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(3*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b*c/d)^2*t
an(b*c/d)^2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d
)*tan(3*b*c/d)^2*tan(b*c/d)^2 - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a) - 6*real_part(cos
_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a) + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2
+ 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*t
an(3*a)^2*tan(3*b*c/d) + 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d) - 2*real_part(cos
_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(3*b*c/d) - 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2*tan(3
*b*c/d) - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(3*b*c/d)^2 - 2*real_part(cos_integral(-6*b*x
 - 6*b*c/d))*tan(3*a)*tan(3*b*c/d)^2 - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(3*b*c/d)^2 - 6*re
al_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(3*b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan
(3*a)^2*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(b*c/d) - 6*real_part(cos_integ
ral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) +
6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*b*c/d)^2*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x - 2*b*c
/d))*tan(3*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(b*c/d)^2 + 2*real_par
t(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*ta
n(b*c/d)^2 + 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 2*real_part(cos_integral(6*b*x
+ 6*b*c/d))*tan(3*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b*c/d)*tan(b*c/d)^2
- imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2 - 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2
+ 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^
2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2 + imag_part(co
s_integral(6*b*x + 6*b*c/d))*tan(a)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 - 3*imag_part(cos_
integral(-2*b*x - 2*b*c/d))*tan(a)^2 - imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2 + 2*sin_integral(6*(
b*d*x + b*c)/d)*tan(a)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2 + 4*imag_part(cos_integral(6*b*x + 6*b*c
/d))*tan(3*a)*tan(3*b*c/d) - 4*imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(3*b*c/d) + 8*sin_integra
l(6*(b*d*x + b*c)/d)*tan(3*a)*tan(3*b*c/d) - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*b*c/d)^2 - 3*imag_
part(cos_integral(2*b*x + 2*b*c/d))*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*b*c/d)^
2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b*c/d)^2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*b*c/d)^
2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*b*c/d)^2 - 12*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(
b*c/d) + 12*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)*t
an(a)*tan(b*c/d) + imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*
b*c/d))*tan(b*c/d)^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - imag_part(cos_integral(-6*b*
x - 6*b*c/d))*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d
)*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a) + 2*real_part(cos_integral(-6*b*x - 6*b*c
/d))*tan(3*a) - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(a) - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*b*c/d) - 2*real_part(cos_integral(-6*b*x - 6*b*c/d
))*tan(3*b*c/d) + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x - 2*
b*c/d))*tan(b*c/d) + imag_part(cos_integral(6*b*x + 6*b*c/d)) - 3*imag_part(cos_integral(2*b*x + 2*b*c/d)) + 3
*imag_part(cos_integral(-2*b*x - 2*b*c/d)) - imag_part(cos_integral(-6*b*x - 6*b*c/d)) + 2*sin_integral(6*(b*d
*x + b*c)/d) - 6*sin_integral(2*(b*d*x + b*c)/d))/(d*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + d*tan(3
*a)^2*tan(a)^2*tan(3*b*c/d)^2 + d*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 + d*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2
+ d*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + d*tan(3*a)^2*tan(a)^2 + d*tan(3*a)^2*tan(3*b*c/d)^2 + d*tan(a)^2*ta
n(3*b*c/d)^2 + d*tan(3*a)^2*tan(b*c/d)^2 + d*tan(a)^2*tan(b*c/d)^2 + d*tan(3*b*c/d)^2*tan(b*c/d)^2 + d*tan(3*a
)^2 + d*tan(a)^2 + d*tan(3*b*c/d)^2 + d*tan(b*c/d)^2 + d)

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