3.28 \(\int \frac{\cos ^4(a+b x)}{x^3} \, dx\)

Optimal. Leaf size=90 \[ -b^2 \cos (2 a) \text{CosIntegral}(2 b x)-b^2 \cos (4 a) \text{CosIntegral}(4 b x)+b^2 \sin (2 a) \text{Si}(2 b x)+b^2 \sin (4 a) \text{Si}(4 b x)-\frac{\cos ^4(a+b x)}{2 x^2}+\frac{2 b \sin (a+b x) \cos ^3(a+b x)}{x} \]

[Out]

-Cos[a + b*x]^4/(2*x^2) - b^2*Cos[2*a]*CosIntegral[2*b*x] - b^2*Cos[4*a]*CosIntegral[4*b*x] + (2*b*Cos[a + b*x
]^3*Sin[a + b*x])/x + b^2*Sin[2*a]*SinIntegral[2*b*x] + b^2*Sin[4*a]*SinIntegral[4*b*x]

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Rubi [A]  time = 0.296895, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3314, 3312, 3303, 3299, 3302} \[ -b^2 \cos (2 a) \text{CosIntegral}(2 b x)-b^2 \cos (4 a) \text{CosIntegral}(4 b x)+b^2 \sin (2 a) \text{Si}(2 b x)+b^2 \sin (4 a) \text{Si}(4 b x)-\frac{\cos ^4(a+b x)}{2 x^2}+\frac{2 b \sin (a+b x) \cos ^3(a+b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4/x^3,x]

[Out]

-Cos[a + b*x]^4/(2*x^2) - b^2*Cos[2*a]*CosIntegral[2*b*x] - b^2*Cos[4*a]*CosIntegral[4*b*x] + (2*b*Cos[a + b*x
]^3*Sin[a + b*x])/x + b^2*Sin[2*a]*SinIntegral[2*b*x] + b^2*Sin[4*a]*SinIntegral[4*b*x]

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && 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^3} \, dx &=-\frac{\cos ^4(a+b x)}{2 x^2}+\frac{2 b \cos ^3(a+b x) \sin (a+b x)}{x}+\left (6 b^2\right ) \int \frac{\cos ^2(a+b x)}{x} \, dx-\left (8 b^2\right ) \int \frac{\cos ^4(a+b x)}{x} \, dx\\ &=-\frac{\cos ^4(a+b x)}{2 x^2}+\frac{2 b \cos ^3(a+b x) \sin (a+b x)}{x}+\left (6 b^2\right ) \int \left (\frac{1}{2 x}+\frac{\cos (2 a+2 b x)}{2 x}\right ) \, dx-\left (8 b^2\right ) \int \left (\frac{3}{8 x}+\frac{\cos (2 a+2 b x)}{2 x}+\frac{\cos (4 a+4 b x)}{8 x}\right ) \, dx\\ &=-\frac{\cos ^4(a+b x)}{2 x^2}+\frac{2 b \cos ^3(a+b x) \sin (a+b x)}{x}-b^2 \int \frac{\cos (4 a+4 b x)}{x} \, dx+\left (3 b^2\right ) \int \frac{\cos (2 a+2 b x)}{x} \, dx-\left (4 b^2\right ) \int \frac{\cos (2 a+2 b x)}{x} \, dx\\ &=-\frac{\cos ^4(a+b x)}{2 x^2}+\frac{2 b \cos ^3(a+b x) \sin (a+b x)}{x}+\left (3 b^2 \cos (2 a)\right ) \int \frac{\cos (2 b x)}{x} \, dx-\left (4 b^2 \cos (2 a)\right ) \int \frac{\cos (2 b x)}{x} \, dx-\left (b^2 \cos (4 a)\right ) \int \frac{\cos (4 b x)}{x} \, dx-\left (3 b^2 \sin (2 a)\right ) \int \frac{\sin (2 b x)}{x} \, dx+\left (4 b^2 \sin (2 a)\right ) \int \frac{\sin (2 b x)}{x} \, dx+\left (b^2 \sin (4 a)\right ) \int \frac{\sin (4 b x)}{x} \, dx\\ &=-\frac{\cos ^4(a+b x)}{2 x^2}-b^2 \cos (2 a) \text{Ci}(2 b x)-b^2 \cos (4 a) \text{Ci}(4 b x)+\frac{2 b \cos ^3(a+b x) \sin (a+b x)}{x}+b^2 \sin (2 a) \text{Si}(2 b x)+b^2 \sin (4 a) \text{Si}(4 b x)\\ \end{align*}

Mathematica [A]  time = 0.302518, size = 119, normalized size = 1.32 \[ -\frac{16 b^2 x^2 \cos (2 a) \text{CosIntegral}(2 b x)+16 b^2 x^2 \cos (4 a) \text{CosIntegral}(4 b x)-16 b^2 x^2 \sin (2 a) \text{Si}(2 b x)-16 b^2 x^2 \sin (4 a) \text{Si}(4 b x)-8 b x \sin (2 (a+b x))-4 b x \sin (4 (a+b x))+4 \cos (2 (a+b x))+\cos (4 (a+b x))+3}{16 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4/x^3,x]

[Out]

-(3 + 4*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 16*b^2*x^2*Cos[2*a]*CosIntegral[2*b*x] + 16*b^2*x^2*Cos[4*a]*Cos
Integral[4*b*x] - 8*b*x*Sin[2*(a + b*x)] - 4*b*x*Sin[4*(a + b*x)] - 16*b^2*x^2*Sin[2*a]*SinIntegral[2*b*x] - 1
6*b^2*x^2*Sin[4*a]*SinIntegral[4*b*x])/(16*x^2)

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Maple [A]  time = 0.033, size = 124, normalized size = 1.4 \begin{align*}{b}^{2} \left ( -{\frac{\cos \left ( 4\,bx+4\,a \right ) }{16\,{x}^{2}{b}^{2}}}+{\frac{\sin \left ( 4\,bx+4\,a \right ) }{4\,bx}}+{\it Si} \left ( 4\,bx \right ) \sin \left ( 4\,a \right ) -{\it Ci} \left ( 4\,bx \right ) \cos \left ( 4\,a \right ) -{\frac{\cos \left ( 2\,bx+2\,a \right ) }{4\,{x}^{2}{b}^{2}}}+{\frac{\sin \left ( 2\,bx+2\,a \right ) }{2\,bx}}+{\it Si} \left ( 2\,bx \right ) \sin \left ( 2\,a \right ) -{\it Ci} \left ( 2\,bx \right ) \cos \left ( 2\,a \right ) -{\frac{3}{16\,{x}^{2}{b}^{2}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^4/x^3,x)

[Out]

b^2*(-1/16*cos(4*b*x+4*a)/x^2/b^2+1/4*sin(4*b*x+4*a)/x/b+Si(4*b*x)*sin(4*a)-Ci(4*b*x)*cos(4*a)-1/4*cos(2*b*x+2
*a)/x^2/b^2+1/2*sin(2*b*x+2*a)/x/b+Si(2*b*x)*sin(2*a)-Ci(2*b*x)*cos(2*a)-3/16/x^2/b^2)

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Maxima [C]  time = 1.38394, size = 1073, normalized size = 11.92 \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^3,x, algorithm="maxima")

[Out]

-1/2097152*(65536*((exp_integral_e(3, 4*I*b*x) + exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(3,
4*I*b*x) + exp_integral_e(3, -4*I*b*x))*sin(2*a)^2)*cos(4*a)^3 - ((65536*I*exp_integral_e(3, 4*I*b*x) - 65536*
I*exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (65536*I*exp_integral_e(3, 4*I*b*x) - 65536*I*exp_integral_e(3, -4
*I*b*x))*sin(2*a)^2)*sin(4*a)^3 + (262144*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a)^
3 - (262144*I*exp_integral_e(3, 2*I*b*x) - 262144*I*exp_integral_e(3, -2*I*b*x))*sin(2*a)^3 + 131072*(2*(exp_i
ntegral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a) + 3)*sin(2*a)^2 + 262144*(exp_integral_e(3, 2*I*
b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a) + 393216*cos(2*a)^2 - ((262144*I*exp_integral_e(3, 2*I*b*x) - 262
144*I*exp_integral_e(3, -2*I*b*x))*cos(2*a)^2 + 262144*I*exp_integral_e(3, 2*I*b*x) - 262144*I*exp_integral_e(
3, -2*I*b*x))*sin(2*a))*cos(4*a)^2 + (262144*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*
a)^3 - (262144*I*exp_integral_e(3, 2*I*b*x) - 262144*I*exp_integral_e(3, -2*I*b*x))*sin(2*a)^3 + 131072*(2*(ex
p_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a) + 3)*sin(2*a)^2 + 65536*((exp_integral_e(3, 4
*I*b*x) + exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(3, 4*I*b*x) + exp_integral_e(3, -4*I*b*x))
*sin(2*a)^2)*cos(4*a) + 262144*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a) + 393216*co
s(2*a)^2 - ((262144*I*exp_integral_e(3, 2*I*b*x) - 262144*I*exp_integral_e(3, -2*I*b*x))*cos(2*a)^2 + 262144*I
*exp_integral_e(3, 2*I*b*x) - 262144*I*exp_integral_e(3, -2*I*b*x))*sin(2*a))*sin(4*a)^2 + 65536*((exp_integra
l_e(3, 4*I*b*x) + exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(3, 4*I*b*x) + exp_integral_e(3, -4
*I*b*x))*sin(2*a)^2)*cos(4*a) - (((65536*I*exp_integral_e(3, 4*I*b*x) - 65536*I*exp_integral_e(3, -4*I*b*x))*c
os(2*a)^2 + (65536*I*exp_integral_e(3, 4*I*b*x) - 65536*I*exp_integral_e(3, -4*I*b*x))*sin(2*a)^2)*cos(4*a)^2
+ (65536*I*exp_integral_e(3, 4*I*b*x) - 65536*I*exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (65536*I*exp_integra
l_e(3, 4*I*b*x) - 65536*I*exp_integral_e(3, -4*I*b*x))*sin(2*a)^2)*sin(4*a))*b^2/(((cos(2*a)^2 + sin(2*a)^2)*c
os(4*a)^2 + (cos(2*a)^2 + sin(2*a)^2)*sin(4*a)^2)*(b*x + a)^2 + (a^2*cos(2*a)^2 + a^2*sin(2*a)^2)*cos(4*a)^2 +
(a^2*cos(2*a)^2 + a^2*sin(2*a)^2)*sin(4*a)^2 - 2*((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)*(b*x + a))

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Fricas [A]  time = 1.44777, size = 389, normalized size = 4.32 \begin{align*} \frac{4 \, b x \cos \left (b x + a\right )^{3} \sin \left (b x + a\right ) + 2 \, b^{2} x^{2} \sin \left (4 \, a\right ) \operatorname{Si}\left (4 \, b x\right ) + 2 \, b^{2} x^{2} \sin \left (2 \, a\right ) \operatorname{Si}\left (2 \, b x\right ) - \cos \left (b x + a\right )^{4} -{\left (b^{2} x^{2} \operatorname{Ci}\left (4 \, b x\right ) + b^{2} x^{2} \operatorname{Ci}\left (-4 \, b x\right )\right )} \cos \left (4 \, a\right ) -{\left (b^{2} x^{2} \operatorname{Ci}\left (2 \, b x\right ) + b^{2} x^{2} \operatorname{Ci}\left (-2 \, b x\right )\right )} \cos \left (2 \, a\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/x^3,x, algorithm="fricas")

[Out]

1/2*(4*b*x*cos(b*x + a)^3*sin(b*x + a) + 2*b^2*x^2*sin(4*a)*sin_integral(4*b*x) + 2*b^2*x^2*sin(2*a)*sin_integ
ral(2*b*x) - cos(b*x + a)^4 - (b^2*x^2*cos_integral(4*b*x) + b^2*x^2*cos_integral(-4*b*x))*cos(4*a) - (b^2*x^2
*cos_integral(2*b*x) + b^2*x^2*cos_integral(-2*b*x))*cos(2*a))/x^2

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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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**4/x**3,x)

[Out]

Integral(cos(a + b*x)**4/x**3, x)

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Giac [C]  time = 1.26472, size = 5292, normalized size = 58.8 \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^3,x, algorithm="giac")

[Out]

1/8*(4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_par
t(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))
*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)
^2*tan(2*a)^2*tan(a)^2 + 8*b^2*x^2*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) -
8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) + 16*b^2*x^2*sin_integral(
2*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b
*x)^2*tan(2*a)*tan(a)^2 - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2
+ 16*b^2*x^2*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(
4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*
tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 4*b^2*x^2*real_par
t(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*
x)^2*tan(b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 4*b^2*x
^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x)
)*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2
+ 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integra
l(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)^
2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos
_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(2*
a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 8*b^2*x^2*imag_part
(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^
2*tan(b*x)^2*tan(2*a) + 16*b^2*x^2*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) + 8*b^2*x^2*imag_part(
cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*t
an(b*x)^2*tan(a) + 16*b^2*x^2*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(a) + 8*b^2*x^2*imag_part(cos_int
egral(2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)
^2*tan(a) + 16*b^2*x^2*sin_integral(2*b*x)*tan(2*b*x)^2*tan(2*a)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(2
*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a) +
16*b^2*x^2*sin_integral(2*b*x)*tan(b*x)^2*tan(2*a)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(2*
b*x)^2*tan(2*a)*tan(a)^2 - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)*tan(a)^2 + 16*b^2*x
^2*sin_integral(4*b*x)*tan(2*b*x)^2*tan(2*a)*tan(a)^2 + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(b*x)^2*ta
n(2*a)*tan(a)^2 - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)*tan(a)^2 + 16*b^2*x^2*sin_inte
gral(4*b*x)*tan(b*x)^2*tan(2*a)*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2 -
4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*t
an(2*b*x)^2*tan(b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2 + 4*b^2*x^2*real_pa
rt(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*
a)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 4*b^2*x^2*real_part(cos_integral(-4
*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*re
al_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(
2*a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2 - 8*b*x*tan(2*b*x)^2*tan(b*x)^2*tan(2
*a)^2*tan(a) - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integr
al(2*b*x))*tan(2*b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(a)^2 - 4*b^2*x^2
*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(b*x)^2*t
an(a)^2 + 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b
*x))*tan(b*x)^2*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(a)^2 - 4*b*x*tan(2*b*x)^2*
tan(b*x)^2*tan(2*a)*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_p
art(cos_integral(2*b*x))*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*a)^2*tan(a)^2 +
4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*a)^2*tan(a)^2 - 8*b*x*tan(2*b*x)^2*tan(b*x)*tan(2*a)^2*tan(a)
^2 - 4*b*x*tan(2*b*x)*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*t
an(2*a) - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a) + 16*b^2*x^2*sin_integral(4*b*x)*tan
(2*b*x)^2*tan(2*a) + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(b*x)^2*tan(2*a) - 8*b^2*x^2*imag_part(cos_in
tegral(-4*b*x))*tan(b*x)^2*tan(2*a) + 16*b^2*x^2*sin_integral(4*b*x)*tan(b*x)^2*tan(2*a) + 8*b^2*x^2*imag_part
(cos_integral(2*b*x))*tan(2*b*x)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(a) + 16
*b^2*x^2*sin_integral(2*b*x)*tan(2*b*x)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(2*b*x))*tan(b*x)^2*tan(a)
- 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(a) + 16*b^2*x^2*sin_integral(2*b*x)*tan(b*x)^2*tan(
a) + 8*b^2*x^2*imag_part(cos_integral(2*b*x))*tan(2*a)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*ta
n(2*a)^2*tan(a) + 16*b^2*x^2*sin_integral(2*b*x)*tan(2*a)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*
tan(2*a)*tan(a)^2 - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*a)*tan(a)^2 + 16*b^2*x^2*sin_integral(4*b*
x)*tan(2*a)*tan(a)^2 - 4*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(4*b*x)
)*tan(2*b*x)^2 - 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b
*x))*tan(2*b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2 - 4*b^2*x^2*real_part(cos_integral(
4*b*x))*tan(b*x)^2 - 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-2
*b*x))*tan(b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(b*x)^2 - 4*b*x*tan(2*b*x)^2*tan(b*x)^2*tan(2
*a) + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*a)^
2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*a)^
2 + 8*b*x*tan(2*b*x)^2*tan(b*x)*tan(2*a)^2 - 4*b*x*tan(2*b*x)*tan(b*x)^2*tan(2*a)^2 - 8*b*x*tan(2*b*x)^2*tan(b
*x)^2*tan(a) + 8*b*x*tan(2*b*x)^2*tan(2*a)^2*tan(a) - 8*b*x*tan(b*x)^2*tan(2*a)^2*tan(a) - 4*b^2*x^2*real_part
(cos_integral(4*b*x))*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(a)^2 + 4*b^2*x^2*real_part(cos_i
ntegral(-2*b*x))*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(a)^2 - 8*b*x*tan(2*b*x)^2*tan(b*x)*t
an(a)^2 + 4*b*x*tan(2*b*x)*tan(b*x)^2*tan(a)^2 - 4*b*x*tan(2*b*x)^2*tan(2*a)*tan(a)^2 + 4*b*x*tan(b*x)^2*tan(2
*a)*tan(a)^2 - 4*b*x*tan(2*b*x)*tan(2*a)^2*tan(a)^2 - 8*b*x*tan(b*x)*tan(2*a)^2*tan(a)^2 + 8*b^2*x^2*imag_part
(cos_integral(4*b*x))*tan(2*a) - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*a) + 16*b^2*x^2*sin_integral(
4*b*x)*tan(2*a) + 8*b^2*x^2*imag_part(cos_integral(2*b*x))*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*
tan(a) + 16*b^2*x^2*sin_integral(2*b*x)*tan(a) + 8*tan(2*b*x)^2*tan(b*x)*tan(2*a)^2*tan(a) - 3*tan(2*b*x)^2*ta
n(b*x)^2*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 - 4*b^2*x^2*r
eal_part(cos_integral(4*b*x)) - 4*b^2*x^2*real_part(cos_integral(2*b*x)) - 4*b^2*x^2*real_part(cos_integral(-2
*b*x)) - 4*b^2*x^2*real_part(cos_integral(-4*b*x)) + 8*b*x*tan(2*b*x)^2*tan(b*x) + 4*b*x*tan(2*b*x)*tan(b*x)^2
- 4*b*x*tan(2*b*x)^2*tan(2*a) + 4*b*x*tan(b*x)^2*tan(2*a) - 4*b*x*tan(2*b*x)*tan(2*a)^2 + 8*b*x*tan(b*x)*tan(
2*a)^2 + 8*b*x*tan(2*b*x)^2*tan(a) - 8*b*x*tan(b*x)^2*tan(a) + 8*b*x*tan(2*a)^2*tan(a) + 4*b*x*tan(2*b*x)*tan(
a)^2 - 8*b*x*tan(b*x)*tan(a)^2 + 4*b*x*tan(2*a)*tan(a)^2 + tan(2*b*x)^2*tan(b*x)^2 + 2*tan(2*b*x)*tan(b*x)^2*t
an(2*a) - 4*tan(2*b*x)^2*tan(2*a)^2 + tan(b*x)^2*tan(2*a)^2 + 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 + 4*b*x*tan(2*b*x) + 8*b*x*tan(b*x) + 4*b*x*tan(2*a) + 8*b*x*tan(a) - 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^2*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + x^2*tan(2*b*x
)^2*tan(b*x)^2*tan(2*a)^2 + x^2*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + x^2*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + x^2*
tan(b*x)^2*tan(2*a)^2*tan(a)^2 + x^2*tan(2*b*x)^2*tan(b*x)^2 + x^2*tan(2*b*x)^2*tan(2*a)^2 + x^2*tan(b*x)^2*ta
n(2*a)^2 + x^2*tan(2*b*x)^2*tan(a)^2 + x^2*tan(b*x)^2*tan(a)^2 + x^2*tan(2*a)^2*tan(a)^2 + x^2*tan(2*b*x)^2 +
x^2*tan(b*x)^2 + x^2*tan(2*a)^2 + x^2*tan(a)^2 + x^2)