3.594 \(\int \frac{\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx\)

Optimal. Leaf size=176 \[ \frac{2}{3} a^3 \text{Si}(2 a x)+\frac{16}{3} a^3 \text{Si}(4 a x)+\frac{\cos ^4(a x)}{a^2 x^5}-\frac{\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}+\frac{a^2}{x}+\frac{32 a^2 \cos ^4(a x)}{3 x}-\frac{10 a^2 \cos ^2(a x)}{x}-\frac{4 \cos ^4(a x)}{3 x^3}+\frac{\cos ^2(a x)}{x^3}+\frac{8 a \sin (a x) \cos ^3(a x)}{3 x^2}-\frac{\sin (a x) \cos ^3(a x)}{a x^4}-\frac{a \sin (a x) \cos (a x)}{x^2} \]

[Out]

a^2/x + Cos[a*x]^2/x^3 - (10*a^2*Cos[a*x]^2)/x + Cos[a*x]^4/(a^2*x^5) - (4*Cos[a*x]^4)/(3*x^3) + (32*a^2*Cos[a
*x]^4)/(3*x) - (a*Cos[a*x]*Sin[a*x])/x^2 - (Cos[a*x]^3*Sin[a*x])/(a*x^4) + (8*a*Cos[a*x]^3*Sin[a*x])/(3*x^2) -
Cos[a*x]^5/(a^2*x^5*(Cos[a*x] + a*x*Sin[a*x])) + (2*a^3*SinIntegral[2*a*x])/3 + (16*a^3*SinIntegral[4*a*x])/3

________________________________________________________________________________________

Rubi [A]  time = 0.299207, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4599, 3314, 30, 3313, 12, 3299} \[ \frac{2}{3} a^3 \text{Si}(2 a x)+\frac{16}{3} a^3 \text{Si}(4 a x)+\frac{\cos ^4(a x)}{a^2 x^5}-\frac{\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}+\frac{a^2}{x}+\frac{32 a^2 \cos ^4(a x)}{3 x}-\frac{10 a^2 \cos ^2(a x)}{x}-\frac{4 \cos ^4(a x)}{3 x^3}+\frac{\cos ^2(a x)}{x^3}+\frac{8 a \sin (a x) \cos ^3(a x)}{3 x^2}-\frac{\sin (a x) \cos ^3(a x)}{a x^4}-\frac{a \sin (a x) \cos (a x)}{x^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a*x]^6/(x^4*(Cos[a*x] + a*x*Sin[a*x])^2),x]

[Out]

a^2/x + Cos[a*x]^2/x^3 - (10*a^2*Cos[a*x]^2)/x + Cos[a*x]^4/(a^2*x^5) - (4*Cos[a*x]^4)/(3*x^3) + (32*a^2*Cos[a
*x]^4)/(3*x) - (a*Cos[a*x]*Sin[a*x])/x^2 - (Cos[a*x]^3*Sin[a*x])/(a*x^4) + (8*a*Cos[a*x]^3*Sin[a*x])/(3*x^2) -
Cos[a*x]^5/(a^2*x^5*(Cos[a*x] + a*x*Sin[a*x])) + (2*a^3*SinIntegral[2*a*x])/3 + (16*a^3*SinIntegral[4*a*x])/3

Rule 4599

Int[(Cos[(a_.)*(x_)]^(n_)*((b_.)*(x_))^(m_))/(Cos[(a_.)*(x_)]*(c_.) + (d_.)*(x_)*Sin[(a_.)*(x_)])^2, x_Symbol]
:> -Simp[(b*(b*x)^(m - 1)*Cos[a*x]^(n - 1))/(a*d*(c*Cos[a*x] + d*x*Sin[a*x])), x] - Dist[(b^2*(n - 1))/d^2, I
nt[(b*x)^(m - 2)*Cos[a*x]^(n - 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, 2 - n]

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin{align*} \int \frac{\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx &=-\frac{\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}-\frac{5 \int \frac{\cos ^4(a x)}{x^6} \, dx}{a^2}\\ &=\frac{\cos ^4(a x)}{a^2 x^5}-\frac{\cos ^3(a x) \sin (a x)}{a x^4}-\frac{\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}-3 \int \frac{\cos ^2(a x)}{x^4} \, dx+4 \int \frac{\cos ^4(a x)}{x^4} \, dx\\ &=\frac{\cos ^2(a x)}{x^3}+\frac{\cos ^4(a x)}{a^2 x^5}-\frac{4 \cos ^4(a x)}{3 x^3}-\frac{a \cos (a x) \sin (a x)}{x^2}-\frac{\cos ^3(a x) \sin (a x)}{a x^4}+\frac{8 a \cos ^3(a x) \sin (a x)}{3 x^2}-\frac{\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}-a^2 \int \frac{1}{x^2} \, dx+\left (2 a^2\right ) \int \frac{\cos ^2(a x)}{x^2} \, dx+\left (8 a^2\right ) \int \frac{\cos ^2(a x)}{x^2} \, dx-\frac{1}{3} \left (32 a^2\right ) \int \frac{\cos ^4(a x)}{x^2} \, dx\\ &=\frac{a^2}{x}+\frac{\cos ^2(a x)}{x^3}-\frac{10 a^2 \cos ^2(a x)}{x}+\frac{\cos ^4(a x)}{a^2 x^5}-\frac{4 \cos ^4(a x)}{3 x^3}+\frac{32 a^2 \cos ^4(a x)}{3 x}-\frac{a \cos (a x) \sin (a x)}{x^2}-\frac{\cos ^3(a x) \sin (a x)}{a x^4}+\frac{8 a \cos ^3(a x) \sin (a x)}{3 x^2}-\frac{\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}+\left (4 a^3\right ) \int -\frac{\sin (2 a x)}{2 x} \, dx+\left (16 a^3\right ) \int -\frac{\sin (2 a x)}{2 x} \, dx-\frac{1}{3} \left (128 a^3\right ) \int \left (-\frac{\sin (2 a x)}{4 x}-\frac{\sin (4 a x)}{8 x}\right ) \, dx\\ &=\frac{a^2}{x}+\frac{\cos ^2(a x)}{x^3}-\frac{10 a^2 \cos ^2(a x)}{x}+\frac{\cos ^4(a x)}{a^2 x^5}-\frac{4 \cos ^4(a x)}{3 x^3}+\frac{32 a^2 \cos ^4(a x)}{3 x}-\frac{a \cos (a x) \sin (a x)}{x^2}-\frac{\cos ^3(a x) \sin (a x)}{a x^4}+\frac{8 a \cos ^3(a x) \sin (a x)}{3 x^2}-\frac{\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}-\left (2 a^3\right ) \int \frac{\sin (2 a x)}{x} \, dx+\frac{1}{3} \left (16 a^3\right ) \int \frac{\sin (4 a x)}{x} \, dx-\left (8 a^3\right ) \int \frac{\sin (2 a x)}{x} \, dx+\frac{1}{3} \left (32 a^3\right ) \int \frac{\sin (2 a x)}{x} \, dx\\ &=\frac{a^2}{x}+\frac{\cos ^2(a x)}{x^3}-\frac{10 a^2 \cos ^2(a x)}{x}+\frac{\cos ^4(a x)}{a^2 x^5}-\frac{4 \cos ^4(a x)}{3 x^3}+\frac{32 a^2 \cos ^4(a x)}{3 x}-\frac{a \cos (a x) \sin (a x)}{x^2}-\frac{\cos ^3(a x) \sin (a x)}{a x^4}+\frac{8 a \cos ^3(a x) \sin (a x)}{3 x^2}-\frac{\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}+\frac{2}{3} a^3 \text{Si}(2 a x)+\frac{16}{3} a^3 \text{Si}(4 a x)\\ \end{align*}

Mathematica [A]  time = 1.2424, size = 194, normalized size = 1.1 \[ \frac{32 a^3 x^3 \text{Si}(2 a x) (a x \sin (a x)+\cos (a x))+256 a^3 x^3 \text{Si}(4 a x) (a x \sin (a x)+\cos (a x))-8 a^3 x^3 \sin (a x)-24 a^3 x^3 \sin (3 a x)+32 a^3 x^3 \sin (5 a x)+12 a^2 x^2 \cos (a x)+44 a^2 x^2 \cos (3 a x)+24 a^2 x^2 \cos (5 a x)+8 a x \sin (a x)+12 a x \sin (3 a x)+4 a x \sin (5 a x)-10 \cos (a x)-5 \cos (3 a x)-\cos (5 a x)}{48 x^3 (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a*x]^6/(x^4*(Cos[a*x] + a*x*Sin[a*x])^2),x]

[Out]

(-10*Cos[a*x] + 12*a^2*x^2*Cos[a*x] - 5*Cos[3*a*x] + 44*a^2*x^2*Cos[3*a*x] - Cos[5*a*x] + 24*a^2*x^2*Cos[5*a*x
] + 8*a*x*Sin[a*x] - 8*a^3*x^3*Sin[a*x] + 12*a*x*Sin[3*a*x] - 24*a^3*x^3*Sin[3*a*x] + 4*a*x*Sin[5*a*x] + 32*a^
3*x^3*Sin[5*a*x] + 32*a^3*x^3*(Cos[a*x] + a*x*Sin[a*x])*SinIntegral[2*a*x] + 256*a^3*x^3*(Cos[a*x] + a*x*Sin[a
*x])*SinIntegral[4*a*x])/(48*x^3*(Cos[a*x] + a*x*Sin[a*x]))

________________________________________________________________________________________

Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( ax \right ) \right ) ^{6}}{{x}^{4} \left ( \cos \left ( ax \right ) +ax\sin \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a*x)^6/x^4/(cos(a*x)+a*x*sin(a*x))^2,x)

[Out]

int(cos(a*x)^6/x^4/(cos(a*x)+a*x*sin(a*x))^2,x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a*x)^6/x^4/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 2.9505, size = 421, normalized size = 2.39 \begin{align*} -\frac{19 \, a^{2} x^{2} \cos \left (a x\right )^{3} -{\left (24 \, a^{2} x^{2} - 1\right )} \cos \left (a x\right )^{5} - 2 \,{\left (8 \, a^{3} x^{3} \operatorname{Si}\left (4 \, a x\right ) + a^{3} x^{3} \operatorname{Si}\left (2 \, a x\right )\right )} \cos \left (a x\right ) -{\left (16 \, a^{4} x^{4} \operatorname{Si}\left (4 \, a x\right ) + 2 \, a^{4} x^{4} \operatorname{Si}\left (2 \, a x\right ) - 30 \, a^{3} x^{3} \cos \left (a x\right )^{2} + 3 \, a^{3} x^{3} + 4 \,{\left (8 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{4}\right )} \sin \left (a x\right )}{3 \,{\left (a x^{4} \sin \left (a x\right ) + x^{3} \cos \left (a x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a*x)^6/x^4/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="fricas")

[Out]

-1/3*(19*a^2*x^2*cos(a*x)^3 - (24*a^2*x^2 - 1)*cos(a*x)^5 - 2*(8*a^3*x^3*sin_integral(4*a*x) + a^3*x^3*sin_int
egral(2*a*x))*cos(a*x) - (16*a^4*x^4*sin_integral(4*a*x) + 2*a^4*x^4*sin_integral(2*a*x) - 30*a^3*x^3*cos(a*x)
^2 + 3*a^3*x^3 + 4*(8*a^3*x^3 + a*x)*cos(a*x)^4)*sin(a*x))/(a*x^4*sin(a*x) + x^3*cos(a*x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{6}{\left (a x \right )}}{x^{4} \left (a x \sin{\left (a x \right )} + \cos{\left (a x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a*x)**6/x**4/(cos(a*x)+a*x*sin(a*x))**2,x)

[Out]

Integral(cos(a*x)**6/(x**4*(a*x*sin(a*x) + cos(a*x))**2), x)

________________________________________________________________________________________

Giac [C]  time = 2.494, size = 9827, normalized size = 55.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a*x)^6/x^4/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="giac")

[Out]

1/12*(64*a^8*x^8*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 8*a^8*x^8*imag_part(cos
_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 8*a^8*x^8*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^
2*tan(a*x)^2*tan(1/2*a*x) - 64*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) +
128*a^8*x^8*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 16*a^8*x^8*sin_integral(2*a*x)*tan(2*a*
x)^2*tan(a*x)^2*tan(1/2*a*x) - 32*a^7*x^7*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^
2 - 4*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^7*x^7*imag_part(cos_
integral(-2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 32*a^7*x^7*imag_part(cos_integral(-4*a*x))*tan(2*a*
x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 64*a^7*x^7*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^7
*x^7*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 64*a^8*x^8*imag_part(cos_integral(4*a*x))*ta
n(2*a*x)^2*tan(1/2*a*x) + 8*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 8*a^8*x^8*imag_
part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 64*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2
*tan(1/2*a*x) + 128*a^8*x^8*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x) + 16*a^8*x^8*sin_integral(2*a*x)*tan
(2*a*x)^2*tan(1/2*a*x) + 64*a^8*x^8*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*x) + 8*a^8*x^8*imag_pa
rt(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 8*a^8*x^8*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2
*a*x) - 64*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x) + 128*a^8*x^8*sin_integral(4*a*x)*t
an(a*x)^2*tan(1/2*a*x) + 16*a^8*x^8*sin_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x) + 32*a^7*x^7*imag_part(cos_int
egral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 4*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 4*a
^7*x^7*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 32*a^7*x^7*imag_part(cos_integral(-4*a*x))*ta
n(2*a*x)^2*tan(a*x)^2 + 64*a^7*x^7*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 8*a^7*x^7*sin_integral(2*a*x)
*tan(2*a*x)^2*tan(a*x)^2 - 40*a^7*x^7*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 32*a^7*x^7*imag_part(cos_integral
(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 4*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 +
4*a^7*x^7*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 32*a^7*x^7*imag_part(cos_integral(-4*a
*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 64*a^7*x^7*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 - 8*a^7*x^7*sin_
integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 - 32*a^7*x^7*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*x
)^2 - 4*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^7*x^7*imag_part(cos_integral(-2
*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 32*a^7*x^7*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 64*a
^7*x^7*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^7*x^7*sin_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^2
+ 64*a^8*x^8*imag_part(cos_integral(4*a*x))*tan(1/2*a*x) + 8*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(1/2*a
*x) - 8*a^8*x^8*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x) - 64*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(
1/2*a*x) + 128*a^8*x^8*sin_integral(4*a*x)*tan(1/2*a*x) + 16*a^8*x^8*sin_integral(2*a*x)*tan(1/2*a*x) + 128*a^
6*x^6*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 16*a^6*x^6*imag_part(cos_integral(
2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 16*a^6*x^6*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(a*x
)^2*tan(1/2*a*x) - 128*a^6*x^6*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 256*a^6*
x^6*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 32*a^6*x^6*sin_integral(2*a*x)*tan(2*a*x)^2*tan
(a*x)^2*tan(1/2*a*x) + 20*a^6*x^6*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 32*a^7*x^7*imag_part(cos_integral(4
*a*x))*tan(2*a*x)^2 + 4*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2 - 4*a^7*x^7*imag_part(cos_integral
(-2*a*x))*tan(2*a*x)^2 - 32*a^7*x^7*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2 + 64*a^7*x^7*sin_integral(4*a
*x)*tan(2*a*x)^2 + 8*a^7*x^7*sin_integral(2*a*x)*tan(2*a*x)^2 + 32*a^7*x^7*imag_part(cos_integral(4*a*x))*tan(
a*x)^2 + 4*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(a*x)^2 - 4*a^7*x^7*imag_part(cos_integral(-2*a*x))*tan(a
*x)^2 - 32*a^7*x^7*imag_part(cos_integral(-4*a*x))*tan(a*x)^2 + 64*a^7*x^7*sin_integral(4*a*x)*tan(a*x)^2 + 8*
a^7*x^7*sin_integral(2*a*x)*tan(a*x)^2 - 24*a^7*x^7*tan(2*a*x)^2*tan(1/2*a*x) + 24*a^7*x^7*tan(a*x)^2*tan(1/2*
a*x) - 32*a^7*x^7*imag_part(cos_integral(4*a*x))*tan(1/2*a*x)^2 - 4*a^7*x^7*imag_part(cos_integral(2*a*x))*tan
(1/2*a*x)^2 + 4*a^7*x^7*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x)^2 + 32*a^7*x^7*imag_part(cos_integral(-4*
a*x))*tan(1/2*a*x)^2 - 64*a^7*x^7*sin_integral(4*a*x)*tan(1/2*a*x)^2 - 8*a^7*x^7*sin_integral(2*a*x)*tan(1/2*a
*x)^2 - 64*a^5*x^5*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^5*x^5*imag_part
(cos_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 8*a^5*x^5*imag_part(cos_integral(-2*a*x))*tan(2
*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 64*a^5*x^5*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2
*a*x)^2 - 128*a^5*x^5*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 16*a^5*x^5*sin_integral(2*a
*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 20*a^6*x^6*tan(2*a*x)^2*tan(a*x)^2 + 128*a^6*x^6*imag_part(cos_in
tegral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 16*a^6*x^6*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)
- 16*a^6*x^6*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 128*a^6*x^6*imag_part(cos_integral(-
4*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 256*a^6*x^6*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x) + 32*a^6*x^6*sin
_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x) + 8*a^6*x^6*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x) + 128*a^6*x^6*imag_p
art(cos_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*x) + 16*a^6*x^6*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/
2*a*x) - 16*a^6*x^6*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 128*a^6*x^6*imag_part(cos_integr
al(-4*a*x))*tan(a*x)^2*tan(1/2*a*x) + 256*a^6*x^6*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x) + 32*a^6*x^6*sin
_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x) + 16*a^6*x^6*tan(2*a*x)*tan(a*x)^2*tan(1/2*a*x) + 12*a^6*x^6*tan(2*a*
x)^2*tan(1/2*a*x)^2 - 12*a^6*x^6*tan(a*x)^2*tan(1/2*a*x)^2 + 32*a^7*x^7*imag_part(cos_integral(4*a*x)) + 4*a^7
*x^7*imag_part(cos_integral(2*a*x)) - 4*a^7*x^7*imag_part(cos_integral(-2*a*x)) - 32*a^7*x^7*imag_part(cos_int
egral(-4*a*x)) + 64*a^7*x^7*sin_integral(4*a*x) + 8*a^7*x^7*sin_integral(2*a*x) + 64*a^5*x^5*imag_part(cos_int
egral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 8*a^5*x^5*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 8*a
^5*x^5*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 64*a^5*x^5*imag_part(cos_integral(-4*a*x))*ta
n(2*a*x)^2*tan(a*x)^2 + 128*a^5*x^5*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 16*a^5*x^5*sin_integral(2*a*
x)*tan(2*a*x)^2*tan(a*x)^2 + 40*a^7*x^7*tan(1/2*a*x) - 72*a^5*x^5*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 64*a^
5*x^5*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 8*a^5*x^5*imag_part(cos_integral(2*a*x))*ta
n(2*a*x)^2*tan(1/2*a*x)^2 + 8*a^5*x^5*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 64*a^5*x^5
*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 128*a^5*x^5*sin_integral(4*a*x)*tan(2*a*x)^2*ta
n(1/2*a*x)^2 - 16*a^5*x^5*sin_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 - 4*a^5*x^5*tan(2*a*x)^2*tan(a*x)*ta
n(1/2*a*x)^2 - 64*a^5*x^5*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^5*x^5*imag_part(cos_i
ntegral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 8*a^5*x^5*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x)^
2 + 64*a^5*x^5*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 128*a^5*x^5*sin_integral(4*a*x)*tan
(a*x)^2*tan(1/2*a*x)^2 - 16*a^5*x^5*sin_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^5*x^5*tan(2*a*x)*tan(a
*x)^2*tan(1/2*a*x)^2 - 12*a^6*x^6*tan(2*a*x)^2 + 12*a^6*x^6*tan(a*x)^2 + 128*a^6*x^6*imag_part(cos_integral(4*
a*x))*tan(1/2*a*x) + 16*a^6*x^6*imag_part(cos_integral(2*a*x))*tan(1/2*a*x) - 16*a^6*x^6*imag_part(cos_integra
l(-2*a*x))*tan(1/2*a*x) - 128*a^6*x^6*imag_part(cos_integral(-4*a*x))*tan(1/2*a*x) + 256*a^6*x^6*sin_integral(
4*a*x)*tan(1/2*a*x) + 32*a^6*x^6*sin_integral(2*a*x)*tan(1/2*a*x) + 16*a^6*x^6*tan(2*a*x)*tan(1/2*a*x) + 8*a^6
*x^6*tan(a*x)*tan(1/2*a*x) + 64*a^4*x^4*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) +
8*a^4*x^4*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 8*a^4*x^4*imag_part(cos_integr
al(-2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 64*a^4*x^4*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan
(a*x)^2*tan(1/2*a*x) + 128*a^4*x^4*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 16*a^4*x^4*sin_i
ntegral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 20*a^6*x^6*tan(1/2*a*x)^2 + 36*a^4*x^4*tan(2*a*x)^2*tan(
a*x)^2*tan(1/2*a*x)^2 + 64*a^5*x^5*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2 + 8*a^5*x^5*imag_part(cos_integ
ral(2*a*x))*tan(2*a*x)^2 - 8*a^5*x^5*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2 - 64*a^5*x^5*imag_part(cos_i
ntegral(-4*a*x))*tan(2*a*x)^2 + 128*a^5*x^5*sin_integral(4*a*x)*tan(2*a*x)^2 + 16*a^5*x^5*sin_integral(2*a*x)*
tan(2*a*x)^2 + 4*a^5*x^5*tan(2*a*x)^2*tan(a*x) + 64*a^5*x^5*imag_part(cos_integral(4*a*x))*tan(a*x)^2 + 8*a^5*
x^5*imag_part(cos_integral(2*a*x))*tan(a*x)^2 - 8*a^5*x^5*imag_part(cos_integral(-2*a*x))*tan(a*x)^2 - 64*a^5*
x^5*imag_part(cos_integral(-4*a*x))*tan(a*x)^2 + 128*a^5*x^5*sin_integral(4*a*x)*tan(a*x)^2 + 16*a^5*x^5*sin_i
ntegral(2*a*x)*tan(a*x)^2 + 8*a^5*x^5*tan(2*a*x)*tan(a*x)^2 - 48*a^5*x^5*tan(2*a*x)^2*tan(1/2*a*x) + 48*a^5*x^
5*tan(a*x)^2*tan(1/2*a*x) - 64*a^5*x^5*imag_part(cos_integral(4*a*x))*tan(1/2*a*x)^2 - 8*a^5*x^5*imag_part(cos
_integral(2*a*x))*tan(1/2*a*x)^2 + 8*a^5*x^5*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x)^2 + 64*a^5*x^5*imag_
part(cos_integral(-4*a*x))*tan(1/2*a*x)^2 - 128*a^5*x^5*sin_integral(4*a*x)*tan(1/2*a*x)^2 - 16*a^5*x^5*sin_in
tegral(2*a*x)*tan(1/2*a*x)^2 - 8*a^5*x^5*tan(2*a*x)*tan(1/2*a*x)^2 - 4*a^5*x^5*tan(a*x)*tan(1/2*a*x)^2 - 32*a^
3*x^3*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 4*a^3*x^3*imag_part(cos_integral
(2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^3*x^3*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(a
*x)^2*tan(1/2*a*x)^2 + 32*a^3*x^3*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 64*
a^3*x^3*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^3*x^3*sin_integral(2*a*x)*tan(2*a*x)^
2*tan(a*x)^2*tan(1/2*a*x)^2 + 20*a^6*x^6 - 36*a^4*x^4*tan(2*a*x)^2*tan(a*x)^2 + 64*a^4*x^4*imag_part(cos_integ
ral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 8*a^4*x^4*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 8
*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 64*a^4*x^4*imag_part(cos_integral(-4*a*x)
)*tan(2*a*x)^2*tan(1/2*a*x) + 128*a^4*x^4*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x) + 16*a^4*x^4*sin_integ
ral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x) + 4*a^4*x^4*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x) + 64*a^4*x^4*imag_part(cos
_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*x) + 8*a^4*x^4*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x) -
8*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 64*a^4*x^4*imag_part(cos_integral(-4*a*x)
)*tan(a*x)^2*tan(1/2*a*x) + 128*a^4*x^4*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x) + 16*a^4*x^4*sin_integral(
2*a*x)*tan(a*x)^2*tan(1/2*a*x) + 26*a^4*x^4*tan(2*a*x)*tan(a*x)^2*tan(1/2*a*x) + 24*a^4*x^4*tan(2*a*x)^2*tan(1
/2*a*x)^2 - 24*a^4*x^4*tan(a*x)^2*tan(1/2*a*x)^2 + 64*a^5*x^5*imag_part(cos_integral(4*a*x)) + 8*a^5*x^5*imag_
part(cos_integral(2*a*x)) - 8*a^5*x^5*imag_part(cos_integral(-2*a*x)) - 64*a^5*x^5*imag_part(cos_integral(-4*a
*x)) + 128*a^5*x^5*sin_integral(4*a*x) + 16*a^5*x^5*sin_integral(2*a*x) + 8*a^5*x^5*tan(2*a*x) + 4*a^5*x^5*tan
(a*x) + 32*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 4*a^3*x^3*imag_part(cos_integral(2
*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 4*a^3*x^3*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 32*a^3*x^
3*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 64*a^3*x^3*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*
x)^2 + 8*a^3*x^3*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 72*a^5*x^5*tan(1/2*a*x) - 30*a^3*x^3*tan(2*a*x)
^2*tan(a*x)^2*tan(1/2*a*x) - 32*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 4*a^3*x^3
*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 4*a^3*x^3*imag_part(cos_integral(-2*a*x))*tan(2*
a*x)^2*tan(1/2*a*x)^2 + 32*a^3*x^3*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 64*a^3*x^3*si
n_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 - 8*a^3*x^3*sin_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 - 2*
a^3*x^3*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x)^2 - 32*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*
x)^2 - 4*a^3*x^3*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^3*x^3*imag_part(cos_integral(-
2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 32*a^3*x^3*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 64*
a^3*x^3*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^3*x^3*sin_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^
2 - 13*a^3*x^3*tan(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 24*a^4*x^4*tan(2*a*x)^2 + 24*a^4*x^4*tan(a*x)^2 + 64*a^4
*x^4*imag_part(cos_integral(4*a*x))*tan(1/2*a*x) + 8*a^4*x^4*imag_part(cos_integral(2*a*x))*tan(1/2*a*x) - 8*a
^4*x^4*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x) - 64*a^4*x^4*imag_part(cos_integral(-4*a*x))*tan(1/2*a*x)
+ 128*a^4*x^4*sin_integral(4*a*x)*tan(1/2*a*x) + 16*a^4*x^4*sin_integral(2*a*x)*tan(1/2*a*x) + 26*a^4*x^4*tan(
2*a*x)*tan(1/2*a*x) + 4*a^4*x^4*tan(a*x)*tan(1/2*a*x) - 36*a^4*x^4*tan(1/2*a*x)^2 + 27*a^2*x^2*tan(2*a*x)^2*ta
n(a*x)^2*tan(1/2*a*x)^2 + 32*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2 + 4*a^3*x^3*imag_part(cos_int
egral(2*a*x))*tan(2*a*x)^2 - 4*a^3*x^3*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2 - 32*a^3*x^3*imag_part(cos
_integral(-4*a*x))*tan(2*a*x)^2 + 64*a^3*x^3*sin_integral(4*a*x)*tan(2*a*x)^2 + 8*a^3*x^3*sin_integral(2*a*x)*
tan(2*a*x)^2 + 2*a^3*x^3*tan(2*a*x)^2*tan(a*x) + 32*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(a*x)^2 + 4*a^3*
x^3*imag_part(cos_integral(2*a*x))*tan(a*x)^2 - 4*a^3*x^3*imag_part(cos_integral(-2*a*x))*tan(a*x)^2 - 32*a^3*
x^3*imag_part(cos_integral(-4*a*x))*tan(a*x)^2 + 64*a^3*x^3*sin_integral(4*a*x)*tan(a*x)^2 + 8*a^3*x^3*sin_int
egral(2*a*x)*tan(a*x)^2 + 13*a^3*x^3*tan(2*a*x)*tan(a*x)^2 - 6*a^3*x^3*tan(2*a*x)^2*tan(1/2*a*x) + 24*a^3*x^3*
tan(a*x)^2*tan(1/2*a*x) - 32*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(1/2*a*x)^2 - 4*a^3*x^3*imag_part(cos_i
ntegral(2*a*x))*tan(1/2*a*x)^2 + 4*a^3*x^3*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x)^2 + 32*a^3*x^3*imag_pa
rt(cos_integral(-4*a*x))*tan(1/2*a*x)^2 - 64*a^3*x^3*sin_integral(4*a*x)*tan(1/2*a*x)^2 - 8*a^3*x^3*sin_integr
al(2*a*x)*tan(1/2*a*x)^2 - 13*a^3*x^3*tan(2*a*x)*tan(1/2*a*x)^2 - 2*a^3*x^3*tan(a*x)*tan(1/2*a*x)^2 + 36*a^4*x
^4 - 27*a^2*x^2*tan(2*a*x)^2*tan(a*x)^2 + 20*a^2*x^2*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x) + 10*a^2*x^2*tan(2*a*x
)*tan(a*x)^2*tan(1/2*a*x) + 15*a^2*x^2*tan(2*a*x)^2*tan(1/2*a*x)^2 + 32*a^3*x^3*imag_part(cos_integral(4*a*x))
+ 4*a^3*x^3*imag_part(cos_integral(2*a*x)) - 4*a^3*x^3*imag_part(cos_integral(-2*a*x)) - 32*a^3*x^3*imag_part
(cos_integral(-4*a*x)) + 64*a^3*x^3*sin_integral(4*a*x) + 8*a^3*x^3*sin_integral(2*a*x) + 13*a^3*x^3*tan(2*a*x
) + 2*a^3*x^3*tan(a*x) + 48*a^3*x^3*tan(1/2*a*x) + 2*a*x*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 10*a*x*tan(2*a
*x)^2*tan(a*x)*tan(1/2*a*x)^2 - 5*a*x*tan(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 15*a^2*x^2*tan(2*a*x)^2 + 10*a^2*
x^2*tan(2*a*x)*tan(1/2*a*x) + 20*a^2*x^2*tan(a*x)*tan(1/2*a*x) - 12*a^2*x^2*tan(1/2*a*x)^2 - tan(2*a*x)^2*tan(
a*x)^2*tan(1/2*a*x)^2 + 10*a*x*tan(2*a*x)^2*tan(a*x) + 5*a*x*tan(2*a*x)*tan(a*x)^2 - 6*a*x*tan(2*a*x)^2*tan(1/
2*a*x) - 5*a*x*tan(2*a*x)*tan(1/2*a*x)^2 - 10*a*x*tan(a*x)*tan(1/2*a*x)^2 + 12*a^2*x^2 + tan(2*a*x)^2*tan(a*x)
^2 + 3*tan(2*a*x)^2*tan(1/2*a*x)^2 + 5*a*x*tan(2*a*x) + 10*a*x*tan(a*x) - 8*a*x*tan(1/2*a*x) - 3*tan(2*a*x)^2
+ 4*tan(1/2*a*x)^2 - 4)/(2*a^5*x^8*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - a^4*x^7*tan(2*a*x)^2*tan(a*x)^2*tan(
1/2*a*x)^2 + 2*a^5*x^8*tan(2*a*x)^2*tan(1/2*a*x) + 2*a^5*x^8*tan(a*x)^2*tan(1/2*a*x) + a^4*x^7*tan(2*a*x)^2*ta
n(a*x)^2 - a^4*x^7*tan(2*a*x)^2*tan(1/2*a*x)^2 - a^4*x^7*tan(a*x)^2*tan(1/2*a*x)^2 + 2*a^5*x^8*tan(1/2*a*x) +
4*a^3*x^6*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + a^4*x^7*tan(2*a*x)^2 + a^4*x^7*tan(a*x)^2 - a^4*x^7*tan(1/2*a
*x)^2 - 2*a^2*x^5*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^3*x^6*tan(2*a*x)^2*tan(1/2*a*x) + 4*a^3*x^6*tan
(a*x)^2*tan(1/2*a*x) + a^4*x^7 + 2*a^2*x^5*tan(2*a*x)^2*tan(a*x)^2 - 2*a^2*x^5*tan(2*a*x)^2*tan(1/2*a*x)^2 - 2
*a^2*x^5*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^3*x^6*tan(1/2*a*x) + 2*a*x^4*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 2
*a^2*x^5*tan(2*a*x)^2 + 2*a^2*x^5*tan(a*x)^2 - 2*a^2*x^5*tan(1/2*a*x)^2 - x^3*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*
a*x)^2 + 2*a*x^4*tan(2*a*x)^2*tan(1/2*a*x) + 2*a*x^4*tan(a*x)^2*tan(1/2*a*x) + 2*a^2*x^5 + x^3*tan(2*a*x)^2*ta
n(a*x)^2 - x^3*tan(2*a*x)^2*tan(1/2*a*x)^2 - x^3*tan(a*x)^2*tan(1/2*a*x)^2 + 2*a*x^4*tan(1/2*a*x) + x^3*tan(2*
a*x)^2 + x^3*tan(a*x)^2 - x^3*tan(1/2*a*x)^2 + x^3)