3.27 \(\int \frac{\cos ^4(a+b x)}{x^2} \, dx\)
Optimal. Leaf size=66 \[ -b \sin (2 a) \text{CosIntegral}(2 b x)-\frac{1}{2} b \sin (4 a) \text{CosIntegral}(4 b x)-b \cos (2 a) \text{Si}(2 b x)-\frac{1}{2} b \cos (4 a) \text{Si}(4 b x)-\frac{\cos ^4(a+b x)}{x} \]
[Out]
-(Cos[a + b*x]^4/x) - b*CosIntegral[2*b*x]*Sin[2*a] - (b*CosIntegral[4*b*x]*Sin[4*a])/2 - b*Cos[2*a]*SinIntegr
al[2*b*x] - (b*Cos[4*a]*SinIntegral[4*b*x])/2
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Rubi [A] time = 0.150996, antiderivative size = 66, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{3313, 3303, 3299, 3302} \[ -b \sin (2 a) \text{CosIntegral}(2 b x)-\frac{1}{2} b \sin (4 a) \text{CosIntegral}(4 b x)-b \cos (2 a) \text{Si}(2 b x)-\frac{1}{2} b \cos (4 a) \text{Si}(4 b x)-\frac{\cos ^4(a+b x)}{x} \]
Antiderivative was successfully verified.
[In]
Int[Cos[a + b*x]^4/x^2,x]
[Out]
-(Cos[a + b*x]^4/x) - b*CosIntegral[2*b*x]*Sin[2*a] - (b*CosIntegral[4*b*x]*Sin[4*a])/2 - b*Cos[2*a]*SinIntegr
al[2*b*x] - (b*Cos[4*a]*SinIntegral[4*b*x])/2
Rule 3313
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]
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 ^4(a+b x)}{x^2} \, dx &=-\frac{\cos ^4(a+b x)}{x}+(4 b) \int \left (-\frac{\sin (2 a+2 b x)}{4 x}-\frac{\sin (4 a+4 b x)}{8 x}\right ) \, dx\\ &=-\frac{\cos ^4(a+b x)}{x}-\frac{1}{2} b \int \frac{\sin (4 a+4 b x)}{x} \, dx-b \int \frac{\sin (2 a+2 b x)}{x} \, dx\\ &=-\frac{\cos ^4(a+b x)}{x}-(b \cos (2 a)) \int \frac{\sin (2 b x)}{x} \, dx-\frac{1}{2} (b \cos (4 a)) \int \frac{\sin (4 b x)}{x} \, dx-(b \sin (2 a)) \int \frac{\cos (2 b x)}{x} \, dx-\frac{1}{2} (b \sin (4 a)) \int \frac{\cos (4 b x)}{x} \, dx\\ &=-\frac{\cos ^4(a+b x)}{x}-b \text{Ci}(2 b x) \sin (2 a)-\frac{1}{2} b \text{Ci}(4 b x) \sin (4 a)-b \cos (2 a) \text{Si}(2 b x)-\frac{1}{2} b \cos (4 a) \text{Si}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.21995, size = 79, normalized size = 1.2 \[ -\frac{8 b x \sin (2 a) \text{CosIntegral}(2 b x)+4 b x \sin (4 a) \text{CosIntegral}(4 b x)+8 b x \cos (2 a) \text{Si}(2 b x)+4 b x \cos (4 a) \text{Si}(4 b x)+4 \cos (2 (a+b x))+\cos (4 (a+b x))+3}{8 x} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[a + b*x]^4/x^2,x]
[Out]
-(3 + 4*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 8*b*x*CosIntegral[2*b*x]*Sin[2*a] + 4*b*x*CosIntegral[4*b*x]*Sin
[4*a] + 8*b*x*Cos[2*a]*SinIntegral[2*b*x] + 4*b*x*Cos[4*a]*SinIntegral[4*b*x])/(8*x)
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Maple [A] time = 0.03, size = 90, normalized size = 1.4 \begin{align*} b \left ( -{\frac{\cos \left ( 4\,bx+4\,a \right ) }{8\,bx}}-{\frac{{\it Si} \left ( 4\,bx \right ) \cos \left ( 4\,a \right ) }{2}}-{\frac{{\it Ci} \left ( 4\,bx \right ) \sin \left ( 4\,a \right ) }{2}}-{\frac{\cos \left ( 2\,bx+2\,a \right ) }{2\,bx}}-{\it Si} \left ( 2\,bx \right ) \cos \left ( 2\,a \right ) -{\it Ci} \left ( 2\,bx \right ) \sin \left ( 2\,a \right ) -{\frac{3}{8\,bx}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(b*x+a)^4/x^2,x)
[Out]
b*(-1/8*cos(4*b*x+4*a)/x/b-1/2*Si(4*b*x)*cos(4*a)-1/2*Ci(4*b*x)*sin(4*a)-1/2*cos(2*b*x+2*a)/x/b-Si(2*b*x)*cos(
2*a)-Ci(2*b*x)*sin(2*a)-3/8/x/b)
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Maxima [C] time = 1.35022, size = 980, normalized size = 14.85 \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)^4/x^2,x, algorithm="maxima")
[Out]
1/1048576*(32768*((exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(2, 4
*I*b*x) + exp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*cos(4*a)^3 - ((32768*I*exp_integral_e(2, 4*I*b*x) - 32768*I
*exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (32768*I*exp_integral_e(2, 4*I*b*x) - 32768*I*exp_integral_e(2, -4*
I*b*x))*sin(2*a)^2)*sin(4*a)^3 + (131072*(exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a)^3
- (131072*I*exp_integral_e(2, 2*I*b*x) - 131072*I*exp_integral_e(2, -2*I*b*x))*sin(2*a)^3 + 131072*((exp_inte
gral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 3)*sin(2*a)^2 + 131072*(exp_integral_e(2, 2*I*b*x
) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 393216*cos(2*a)^2 - ((131072*I*exp_integral_e(2, 2*I*b*x) - 131072
*I*exp_integral_e(2, -2*I*b*x))*cos(2*a)^2 + 131072*I*exp_integral_e(2, 2*I*b*x) - 131072*I*exp_integral_e(2,
-2*I*b*x))*sin(2*a))*cos(4*a)^2 + (131072*(exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a)^
3 - (131072*I*exp_integral_e(2, 2*I*b*x) - 131072*I*exp_integral_e(2, -2*I*b*x))*sin(2*a)^3 + 131072*((exp_int
egral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 3)*sin(2*a)^2 + 32768*((exp_integral_e(2, 4*I*b*
x) + exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*sin(
2*a)^2)*cos(4*a) + 131072*(exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 393216*cos(2*a
)^2 - ((131072*I*exp_integral_e(2, 2*I*b*x) - 131072*I*exp_integral_e(2, -2*I*b*x))*cos(2*a)^2 + 131072*I*exp_
integral_e(2, 2*I*b*x) - 131072*I*exp_integral_e(2, -2*I*b*x))*sin(2*a))*sin(4*a)^2 + 32768*((exp_integral_e(2
, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*
x))*sin(2*a)^2)*cos(4*a) - (((32768*I*exp_integral_e(2, 4*I*b*x) - 32768*I*exp_integral_e(2, -4*I*b*x))*cos(2*
a)^2 + (32768*I*exp_integral_e(2, 4*I*b*x) - 32768*I*exp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*cos(4*a)^2 + (32
768*I*exp_integral_e(2, 4*I*b*x) - 32768*I*exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (32768*I*exp_integral_e(2
, 4*I*b*x) - 32768*I*exp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*sin(4*a))*b/((a*cos(2*a)^2 + a*sin(2*a)^2)*cos(4
*a)^2 + (a*cos(2*a)^2 + a*sin(2*a)^2)*sin(4*a)^2 - ((cos(2*a)^2 + sin(2*a)^2)*cos(4*a)^2 + (cos(2*a)^2 + sin(2
*a)^2)*sin(4*a)^2)*(b*x + a))
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Fricas [A] time = 1.44193, size = 312, normalized size = 4.73 \begin{align*} -\frac{4 \, \cos \left (b x + a\right )^{4} + 2 \, b x \cos \left (4 \, a\right ) \operatorname{Si}\left (4 \, b x\right ) + 4 \, b x \cos \left (2 \, a\right ) \operatorname{Si}\left (2 \, b x\right ) +{\left (b x \operatorname{Ci}\left (4 \, b x\right ) + b x \operatorname{Ci}\left (-4 \, b x\right )\right )} \sin \left (4 \, a\right ) + 2 \,{\left (b x \operatorname{Ci}\left (2 \, b x\right ) + b x \operatorname{Ci}\left (-2 \, b x\right )\right )} \sin \left (2 \, a\right )}{4 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^4/x^2,x, algorithm="fricas")
[Out]
-1/4*(4*cos(b*x + a)^4 + 2*b*x*cos(4*a)*sin_integral(4*b*x) + 4*b*x*cos(2*a)*sin_integral(2*b*x) + (b*x*cos_in
tegral(4*b*x) + b*x*cos_integral(-4*b*x))*sin(4*a) + 2*(b*x*cos_integral(2*b*x) + b*x*cos_integral(-2*b*x))*si
n(2*a))/x
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{4}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)**4/x**2,x)
[Out]
Integral(cos(a + b*x)**4/x**2, x)
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Giac [C] time = 1.25907, size = 4347, normalized size = 65.86 \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)^4/x^2,x, algorithm="giac")
[Out]
1/4*(b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 2*b*x*imag_part(cos_inte
gral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*
tan(b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)
^2 + 2*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*b
*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 4*b*x*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2
*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) - 2*b*x*real_part(co
s_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 - 2*b*x*real_part(cos_integral(-4*b*x))*tan(2*b*x
)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 + b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 2*b
*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 2*b*x*imag_part(cos_integral(-2*b*x))*t
an(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 2
*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^
2*tan(2*a)^2 - b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 2*b*x*imag_part(cos_integ
ral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*t
an(a)^2 + b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 - 2*b*x*sin_integral(4*b*x)*tan
(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + b*x*imag_part(cos
_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*a
)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_inte
gral(-4*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 2*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 +
4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + b*x*imag_part(cos_integral(4*b*x))*tan(b*x)^2*tan
(2*a)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 2*b*x*imag_part(cos_i
ntegral(-2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2*ta
n(a)^2 + 2*b*x*sin_integral(4*b*x)*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(b*x)^2*tan(2
*a)^2*tan(a)^2 - 2*b*x*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) - 2*b*x*real_part(cos_i
ntegral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) - 4*b*x*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)
^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a) - 4*b*x*real_part(cos_integra
l(2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a
) - 4*b*x*real_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*
tan(b*x)^2*tan(2*a)^2*tan(a) - 2*b*x*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)*tan(a)^2 - 2*b*x*rea
l_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)*tan(a)^2 - 2*b*x*real_part(cos_integral(4*b*x))*tan(b*x)^2*
tan(2*a)*tan(a)^2 - 2*b*x*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)*tan(a)^2 - 4*tan(2*b*x)^2*tan(b*
x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2 - 2*b*x*imag_part(cos_in
tegral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2 + 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2 + b*x*i
mag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2 - 2*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2 - 4
*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^2 + b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2
- 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b
*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 2*b*x*sin_integral(4*b*x)*tan
(2*b*x)^2*tan(2*a)^2 - 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(2*a)^2 + b*x*imag_part(cos_integral(4*b*x))*
tan(b*x)^2*tan(2*a)^2 - 2*b*x*imag_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2 + 2*b*x*imag_part(cos_integ
ral(-2*b*x))*tan(b*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2 + 2*b*x*sin_int
egral(4*b*x)*tan(b*x)^2*tan(2*a)^2 - 4*b*x*sin_integral(2*b*x)*tan(b*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integ
ral(4*b*x))*tan(2*b*x)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(a)^2 - 2*b*x*imag_pa
rt(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(a)^2 + b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(a)^2 - 2
*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(a)^2 - b*x*imag_pa
rt(cos_integral(4*b*x))*tan(b*x)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(b*x)^2*tan(a)^2 - 2*b*x
*imag_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(a)^2 + b*x*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(a)^2
- 2*b*x*sin_integral(4*b*x)*tan(b*x)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(b*x)^2*tan(a)^2 + b*x*imag_pa
rt(cos_integral(4*b*x))*tan(2*a)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*a)^2*tan(a)^2 - 2*b*x
*imag_part(cos_integral(-2*b*x))*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*a)^2*tan(a)^2
+ 2*b*x*sin_integral(4*b*x)*tan(2*a)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*a)^2*tan(a)^2 - 2*b*x*real_
part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a) - 2*b*x*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)
- 2*b*x*real_part(cos_integral(4*b*x))*tan(b*x)^2*tan(2*a) - 2*b*x*real_part(cos_integral(-4*b*x))*tan(b*x)^2*
tan(2*a) - 4*b*x*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*ta
n(2*b*x)^2*tan(a) - 4*b*x*real_part(cos_integral(2*b*x))*tan(b*x)^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b
*x))*tan(b*x)^2*tan(a) - 4*b*x*real_part(cos_integral(2*b*x))*tan(2*a)^2*tan(a) - 4*b*x*real_part(cos_integral
(-2*b*x))*tan(2*a)^2*tan(a) + 8*tan(2*b*x)^2*tan(b*x)*tan(2*a)^2*tan(a) - 3*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 -
2*b*x*real_part(cos_integral(4*b*x))*tan(2*a)*tan(a)^2 - 2*b*x*real_part(cos_integral(-4*b*x))*tan(2*a)*tan(a
)^2 + 2*tan(2*b*x)*tan(b*x)^2*tan(2*a)*tan(a)^2 - 3*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integra
l(4*b*x))*tan(2*b*x)^2 - 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2 + 2*b*x*imag_part(cos_integral(-2*b
*x))*tan(2*b*x)^2 + b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2 - 2*b*x*sin_integral(4*b*x)*tan(2*b*x)^2
- 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2 - b*x*imag_part(cos_integral(4*b*x))*tan(b*x)^2 - 2*b*x*imag_part(cos
_integral(2*b*x))*tan(b*x)^2 + 2*b*x*imag_part(cos_integral(-2*b*x))*tan(b*x)^2 + b*x*imag_part(cos_integral(-
4*b*x))*tan(b*x)^2 - 2*b*x*sin_integral(4*b*x)*tan(b*x)^2 - 4*b*x*sin_integral(2*b*x)*tan(b*x)^2 + b*x*imag_pa
rt(cos_integral(4*b*x))*tan(2*a)^2 - 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*a)^2 + 2*b*x*imag_part(cos_int
egral(-2*b*x))*tan(2*a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*a)^2 + 2*b*x*sin_integral(4*b*x)*tan(2*a
)^2 - 4*b*x*sin_integral(2*b*x)*tan(2*a)^2 - b*x*imag_part(cos_integral(4*b*x))*tan(a)^2 + 2*b*x*imag_part(cos
_integral(2*b*x))*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(a)^2 + b*x*imag_part(cos_integral(-4*b*
x))*tan(a)^2 - 2*b*x*sin_integral(4*b*x)*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(a)^2 + tan(2*b*x)^2*tan(b*x)
^2 - 2*b*x*real_part(cos_integral(4*b*x))*tan(2*a) - 2*b*x*real_part(cos_integral(-4*b*x))*tan(2*a) + 2*tan(2*
b*x)*tan(b*x)^2*tan(2*a) - 4*tan(2*b*x)^2*tan(2*a)^2 + tan(b*x)^2*tan(2*a)^2 - 4*b*x*real_part(cos_integral(2*
b*x))*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*tan(a) + 8*tan(2*b*x)^2*tan(b*x)*tan(a) + 8*tan(b*x)*tan(
2*a)^2*tan(a) + tan(2*b*x)^2*tan(a)^2 - 4*tan(b*x)^2*tan(a)^2 + 2*tan(2*b*x)*tan(2*a)*tan(a)^2 + tan(2*a)^2*ta
n(a)^2 - b*x*imag_part(cos_integral(4*b*x)) - 2*b*x*imag_part(cos_integral(2*b*x)) + 2*b*x*imag_part(cos_integ
ral(-2*b*x)) + b*x*imag_part(cos_integral(-4*b*x)) - 2*b*x*sin_integral(4*b*x) - 4*b*x*sin_integral(2*b*x) - 3
*tan(2*b*x)^2 + 2*tan(2*b*x)*tan(2*a) - 3*tan(2*a)^2 + 8*tan(b*x)*tan(a) - 4)/(x*tan(2*b*x)^2*tan(b*x)^2*tan(2
*a)^2*tan(a)^2 + x*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + x*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + x*tan(2*b*x)^2*ta
n(2*a)^2*tan(a)^2 + x*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + x*tan(2*b*x)^2*tan(b*x)^2 + x*tan(2*b*x)^2*tan(2*a)^2 +
x*tan(b*x)^2*tan(2*a)^2 + x*tan(2*b*x)^2*tan(a)^2 + x*tan(b*x)^2*tan(a)^2 + x*tan(2*a)^2*tan(a)^2 + x*tan(2*b
*x)^2 + x*tan(b*x)^2 + x*tan(2*a)^2 + x*tan(a)^2 + x)