3.595 \(\int \frac{\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx\)

Optimal. Leaf size=132 \[ -\frac{1}{8} a^2 \text{CosIntegral}(a x)-\frac{27}{8} a^2 \text{CosIntegral}(3 a x)+\frac{\cos ^3(a x)}{a^2 x^4}-\frac{\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}-\frac{3 \cos ^3(a x)}{2 x^2}+\frac{\cos (a x)}{x^2}-\frac{\sin (a x) \cos ^2(a x)}{a x^3}-\frac{a \sin (a x)}{x}+\frac{9 a \sin (a x) \cos ^2(a x)}{2 x} \]

[Out]

Cos[a*x]/x^2 + Cos[a*x]^3/(a^2*x^4) - (3*Cos[a*x]^3)/(2*x^2) - (a^2*CosIntegral[a*x])/8 - (27*a^2*CosIntegral[
3*a*x])/8 - (a*Sin[a*x])/x - (Cos[a*x]^2*Sin[a*x])/(a*x^3) + (9*a*Cos[a*x]^2*Sin[a*x])/(2*x) - Cos[a*x]^4/(a^2
*x^4*(Cos[a*x] + a*x*Sin[a*x]))

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Rubi [A]  time = 0.226077, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4599, 3314, 3297, 3302, 3312} \[ -\frac{1}{8} a^2 \text{CosIntegral}(a x)-\frac{27}{8} a^2 \text{CosIntegral}(3 a x)+\frac{\cos ^3(a x)}{a^2 x^4}-\frac{\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}-\frac{3 \cos ^3(a x)}{2 x^2}+\frac{\cos (a x)}{x^2}-\frac{\sin (a x) \cos ^2(a x)}{a x^3}-\frac{a \sin (a x)}{x}+\frac{9 a \sin (a x) \cos ^2(a x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a*x]^5/(x^3*(Cos[a*x] + a*x*Sin[a*x])^2),x]

[Out]

Cos[a*x]/x^2 + Cos[a*x]^3/(a^2*x^4) - (3*Cos[a*x]^3)/(2*x^2) - (a^2*CosIntegral[a*x])/8 - (27*a^2*CosIntegral[
3*a*x])/8 - (a*Sin[a*x])/x - (Cos[a*x]^2*Sin[a*x])/(a*x^3) + (9*a*Cos[a*x]^2*Sin[a*x])/(2*x) - Cos[a*x]^4/(a^2
*x^4*(Cos[a*x] + a*x*Sin[a*x]))

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 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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]

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])
)

Rubi steps

\begin{align*} \int \frac{\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx &=-\frac{\cos ^4(a x)}{a^2 x^4 (\cos (a x)+a x \sin (a x))}-\frac{4 \int \frac{\cos ^3(a x)}{x^5} \, dx}{a^2}\\ &=\frac{\cos ^3(a x)}{a^2 x^4}-\frac{\cos ^2(a x) \sin (a x)}{a x^3}-\frac{\cos ^4(a x)}{a^2 x^4 (\cos (a x)+a x \sin (a x))}-2 \int \frac{\cos (a x)}{x^3} \, dx+3 \int \frac{\cos ^3(a x)}{x^3} \, dx\\ &=\frac{\cos (a x)}{x^2}+\frac{\cos ^3(a x)}{a^2 x^4}-\frac{3 \cos ^3(a x)}{2 x^2}-\frac{\cos ^2(a x) \sin (a x)}{a x^3}+\frac{9 a \cos ^2(a x) \sin (a x)}{2 x}-\frac{\cos ^4(a x)}{a^2 x^4 (\cos (a x)+a x \sin (a x))}+a \int \frac{\sin (a x)}{x^2} \, dx+\left (9 a^2\right ) \int \frac{\cos (a x)}{x} \, dx-\frac{1}{2} \left (27 a^2\right ) \int \frac{\cos ^3(a x)}{x} \, dx\\ &=\frac{\cos (a x)}{x^2}+\frac{\cos ^3(a x)}{a^2 x^4}-\frac{3 \cos ^3(a x)}{2 x^2}+9 a^2 \text{Ci}(a x)-\frac{a \sin (a x)}{x}-\frac{\cos ^2(a x) \sin (a x)}{a x^3}+\frac{9 a \cos ^2(a x) \sin (a x)}{2 x}-\frac{\cos ^4(a x)}{a^2 x^4 (\cos (a x)+a x \sin (a x))}+a^2 \int \frac{\cos (a x)}{x} \, dx-\frac{1}{2} \left (27 a^2\right ) \int \left (\frac{3 \cos (a x)}{4 x}+\frac{\cos (3 a x)}{4 x}\right ) \, dx\\ &=\frac{\cos (a x)}{x^2}+\frac{\cos ^3(a x)}{a^2 x^4}-\frac{3 \cos ^3(a x)}{2 x^2}+10 a^2 \text{Ci}(a x)-\frac{a \sin (a x)}{x}-\frac{\cos ^2(a x) \sin (a x)}{a x^3}+\frac{9 a \cos ^2(a x) \sin (a x)}{2 x}-\frac{\cos ^4(a x)}{a^2 x^4 (\cos (a x)+a x \sin (a x))}-\frac{1}{8} \left (27 a^2\right ) \int \frac{\cos (3 a x)}{x} \, dx-\frac{1}{8} \left (81 a^2\right ) \int \frac{\cos (a x)}{x} \, dx\\ &=\frac{\cos (a x)}{x^2}+\frac{\cos ^3(a x)}{a^2 x^4}-\frac{3 \cos ^3(a x)}{2 x^2}-\frac{1}{8} a^2 \text{Ci}(a x)-\frac{27}{8} a^2 \text{Ci}(3 a x)-\frac{a \sin (a x)}{x}-\frac{\cos ^2(a x) \sin (a x)}{a x^3}+\frac{9 a \cos ^2(a x) \sin (a x)}{2 x}-\frac{\cos ^4(a x)}{a^2 x^4 (\cos (a x)+a x \sin (a x))}\\ \end{align*}

Mathematica [A]  time = 0.808307, size = 136, normalized size = 1.03 \[ -\frac{2 a^2 x^2 \text{CosIntegral}(a x) (a x \sin (a x)+\cos (a x))+54 a^2 x^2 \text{CosIntegral}(3 a x) (a x \sin (a x)+\cos (a x))-a^2 x^2-8 a^2 x^2 \cos (2 a x)+9 a^2 x^2 \cos (4 a x)-12 a x \sin (2 a x)-6 a x \sin (4 a x)+4 \cos (2 a x)+\cos (4 a x)+3}{16 x^2 (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a*x]^5/(x^3*(Cos[a*x] + a*x*Sin[a*x])^2),x]

[Out]

-(3 - a^2*x^2 + 4*Cos[2*a*x] - 8*a^2*x^2*Cos[2*a*x] + Cos[4*a*x] + 9*a^2*x^2*Cos[4*a*x] + 2*a^2*x^2*CosIntegra
l[a*x]*(Cos[a*x] + a*x*Sin[a*x]) + 54*a^2*x^2*CosIntegral[3*a*x]*(Cos[a*x] + a*x*Sin[a*x]) - 12*a*x*Sin[2*a*x]
- 6*a*x*Sin[4*a*x])/(16*x^2*(Cos[a*x] + a*x*Sin[a*x]))

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( ax \right ) \right ) ^{5}}{{x}^{3} \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)^5/x^3/(cos(a*x)+a*x*sin(a*x))^2,x)

[Out]

int(cos(a*x)^5/x^3/(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)^5/x^3/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.59391, size = 535, normalized size = 4.05 \begin{align*} \frac{88 \, a^{2} x^{2} \cos \left (a x\right )^{2} - 8 \,{\left (9 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right )^{4} - 16 \, a^{2} x^{2} -{\left (27 \, a^{2} x^{2} \operatorname{Ci}\left (3 \, a x\right ) + a^{2} x^{2} \operatorname{Ci}\left (a x\right ) + a^{2} x^{2} \operatorname{Ci}\left (-a x\right ) + 27 \, a^{2} x^{2} \operatorname{Ci}\left (-3 \, a x\right )\right )} \cos \left (a x\right ) -{\left (27 \, a^{3} x^{3} \operatorname{Ci}\left (3 \, a x\right ) + a^{3} x^{3} \operatorname{Ci}\left (a x\right ) + a^{3} x^{3} \operatorname{Ci}\left (-a x\right ) + 27 \, a^{3} x^{3} \operatorname{Ci}\left (-3 \, a x\right ) - 48 \, a x \cos \left (a x\right )^{3}\right )} \sin \left (a x\right )}{16 \,{\left (a x^{3} \sin \left (a x\right ) + x^{2} \cos \left (a x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a*x)^5/x^3/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="fricas")

[Out]

1/16*(88*a^2*x^2*cos(a*x)^2 - 8*(9*a^2*x^2 + 1)*cos(a*x)^4 - 16*a^2*x^2 - (27*a^2*x^2*cos_integral(3*a*x) + a^
2*x^2*cos_integral(a*x) + a^2*x^2*cos_integral(-a*x) + 27*a^2*x^2*cos_integral(-3*a*x))*cos(a*x) - (27*a^3*x^3
*cos_integral(3*a*x) + a^3*x^3*cos_integral(a*x) + a^3*x^3*cos_integral(-a*x) + 27*a^3*x^3*cos_integral(-3*a*x
) - 48*a*x*cos(a*x)^3)*sin(a*x))/(a*x^3*sin(a*x) + x^2*cos(a*x))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a*x)**5/x**3/(cos(a*x)+a*x*sin(a*x))**2,x)

[Out]

Timed out

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Giac [C]  time = 1.73133, size = 4226, normalized size = 32.02 \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)^5/x^3/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="giac")

[Out]

-1/16*(54*a^7*x^7*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 2*a^7*x^7*real_part(cos_integ
ral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 2*a^7*x^7*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^
3 + 54*a^7*x^7*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 27*a^6*x^6*real_part(cos_integr
al(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - a^6*x^6*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4
- a^6*x^6*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 27*a^6*x^6*real_part(cos_integral(-3*a
*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 54*a^7*x^7*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 2
*a^7*x^7*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^7*x^7*real_part(cos_integral(-a*x))*ta
n(3/2*a*x)^2*tan(1/2*a*x) + 54*a^7*x^7*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 54*a^7*x^
7*real_part(cos_integral(3*a*x))*tan(1/2*a*x)^3 + 2*a^7*x^7*real_part(cos_integral(a*x))*tan(1/2*a*x)^3 + 2*a^
7*x^7*real_part(cos_integral(-a*x))*tan(1/2*a*x)^3 + 54*a^7*x^7*real_part(cos_integral(-3*a*x))*tan(1/2*a*x)^3
- 27*a^6*x^6*real_part(cos_integral(3*a*x))*tan(1/2*a*x)^4 - a^6*x^6*real_part(cos_integral(a*x))*tan(1/2*a*x
)^4 - a^6*x^6*real_part(cos_integral(-a*x))*tan(1/2*a*x)^4 - 27*a^6*x^6*real_part(cos_integral(-3*a*x))*tan(1/
2*a*x)^4 + 54*a^7*x^7*real_part(cos_integral(3*a*x))*tan(1/2*a*x) + 2*a^7*x^7*real_part(cos_integral(a*x))*tan
(1/2*a*x) + 2*a^7*x^7*real_part(cos_integral(-a*x))*tan(1/2*a*x) + 54*a^7*x^7*real_part(cos_integral(-3*a*x))*
tan(1/2*a*x) - 8*a^6*x^6*tan(3/2*a*x)^2*tan(1/2*a*x)^2 - 72*a^6*x^6*tan(3/2*a*x)*tan(1/2*a*x)^3 + 108*a^5*x^5*
real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 4*a^5*x^5*real_part(cos_integral(a*x))*tan(3/2*
a*x)^2*tan(1/2*a*x)^3 + 4*a^5*x^5*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 108*a^5*x^5*re
al_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 27*a^6*x^6*real_part(cos_integral(3*a*x))*tan(3/
2*a*x)^2 + a^6*x^6*real_part(cos_integral(a*x))*tan(3/2*a*x)^2 + a^6*x^6*real_part(cos_integral(-a*x))*tan(3/2
*a*x)^2 + 27*a^6*x^6*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2 - 12*a^5*x^5*tan(3/2*a*x)^2*tan(1/2*a*x)^3
+ 36*a^5*x^5*tan(3/2*a*x)*tan(1/2*a*x)^4 - 54*a^4*x^4*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a
*x)^4 - 2*a^4*x^4*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 2*a^4*x^4*real_part(cos_integra
l(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 54*a^4*x^4*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x
)^4 - 72*a^6*x^6*tan(3/2*a*x)*tan(1/2*a*x) + 108*a^5*x^5*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2
*a*x) + 4*a^5*x^5*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 4*a^5*x^5*real_part(cos_integral(
-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 108*a^5*x^5*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) -
8*a^6*x^6*tan(1/2*a*x)^2 + 108*a^5*x^5*real_part(cos_integral(3*a*x))*tan(1/2*a*x)^3 + 4*a^5*x^5*real_part(co
s_integral(a*x))*tan(1/2*a*x)^3 + 4*a^5*x^5*real_part(cos_integral(-a*x))*tan(1/2*a*x)^3 + 108*a^5*x^5*real_pa
rt(cos_integral(-3*a*x))*tan(1/2*a*x)^3 + 8*a^4*x^4*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 27*a^6*x^6*real_part(cos_i
ntegral(3*a*x)) + a^6*x^6*real_part(cos_integral(a*x)) + a^6*x^6*real_part(cos_integral(-a*x)) + 27*a^6*x^6*re
al_part(cos_integral(-3*a*x)) - 12*a^5*x^5*tan(3/2*a*x)^2*tan(1/2*a*x) + 12*a^5*x^5*tan(1/2*a*x)^3 - 54*a^4*x^
4*real_part(cos_integral(3*a*x))*tan(1/2*a*x)^4 - 2*a^4*x^4*real_part(cos_integral(a*x))*tan(1/2*a*x)^4 - 2*a^
4*x^4*real_part(cos_integral(-a*x))*tan(1/2*a*x)^4 - 54*a^4*x^4*real_part(cos_integral(-3*a*x))*tan(1/2*a*x)^4
+ 108*a^5*x^5*real_part(cos_integral(3*a*x))*tan(1/2*a*x) + 4*a^5*x^5*real_part(cos_integral(a*x))*tan(1/2*a*
x) + 4*a^5*x^5*real_part(cos_integral(-a*x))*tan(1/2*a*x) + 108*a^5*x^5*real_part(cos_integral(-3*a*x))*tan(1/
2*a*x) - 4*a^4*x^4*tan(3/2*a*x)^2*tan(1/2*a*x)^2 - 128*a^4*x^4*tan(3/2*a*x)*tan(1/2*a*x)^3 + 54*a^3*x^3*real_p
art(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 2*a^3*x^3*real_part(cos_integral(a*x))*tan(3/2*a*x)^2
*tan(1/2*a*x)^3 + 2*a^3*x^3*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 54*a^3*x^3*real_part
(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 4*a^4*x^4*tan(1/2*a*x)^4 - 36*a^5*x^5*tan(3/2*a*x) + 54
*a^4*x^4*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2 + 2*a^4*x^4*real_part(cos_integral(a*x))*tan(3/2*a*x)^2
+ 2*a^4*x^4*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2 + 54*a^4*x^4*real_part(cos_integral(-3*a*x))*tan(3/2
*a*x)^2 + 12*a^5*x^5*tan(1/2*a*x) + 64*a^3*x^3*tan(3/2*a*x)*tan(1/2*a*x)^4 - 27*a^2*x^2*real_part(cos_integral
(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - a^2*x^2*real_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 -
a^2*x^2*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 27*a^2*x^2*real_part(cos_integral(-3*a*x
))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 4*a^4*x^4*tan(3/2*a*x)^2 - 128*a^4*x^4*tan(3/2*a*x)*tan(1/2*a*x) + 54*a^3*x
^3*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^3*x^3*real_part(cos_integral(a*x))*tan(3/2
*a*x)^2*tan(1/2*a*x) + 2*a^3*x^3*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 54*a^3*x^3*real_p
art(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) - 4*a^4*x^4*tan(1/2*a*x)^2 + 54*a^3*x^3*real_part(cos_in
tegral(3*a*x))*tan(1/2*a*x)^3 + 2*a^3*x^3*real_part(cos_integral(a*x))*tan(1/2*a*x)^3 + 2*a^3*x^3*real_part(co
s_integral(-a*x))*tan(1/2*a*x)^3 + 54*a^3*x^3*real_part(cos_integral(-3*a*x))*tan(1/2*a*x)^3 + 32*a^2*x^2*tan(
3/2*a*x)^2*tan(1/2*a*x)^4 + 54*a^4*x^4*real_part(cos_integral(3*a*x)) + 2*a^4*x^4*real_part(cos_integral(a*x))
+ 2*a^4*x^4*real_part(cos_integral(-a*x)) + 54*a^4*x^4*real_part(cos_integral(-3*a*x)) - 32*a^3*x^3*tan(3/2*a
*x)^2*tan(1/2*a*x) + 32*a^3*x^3*tan(1/2*a*x)^3 - 27*a^2*x^2*real_part(cos_integral(3*a*x))*tan(1/2*a*x)^4 - a^
2*x^2*real_part(cos_integral(a*x))*tan(1/2*a*x)^4 - a^2*x^2*real_part(cos_integral(-a*x))*tan(1/2*a*x)^4 - 27*
a^2*x^2*real_part(cos_integral(-3*a*x))*tan(1/2*a*x)^4 + 8*a^4*x^4 + 54*a^3*x^3*real_part(cos_integral(3*a*x))
*tan(1/2*a*x) + 2*a^3*x^3*real_part(cos_integral(a*x))*tan(1/2*a*x) + 2*a^3*x^3*real_part(cos_integral(-a*x))*
tan(1/2*a*x) + 54*a^3*x^3*real_part(cos_integral(-3*a*x))*tan(1/2*a*x) + 24*a^2*x^2*tan(3/2*a*x)^2*tan(1/2*a*x
)^2 - 56*a^2*x^2*tan(3/2*a*x)*tan(1/2*a*x)^3 + 16*a^2*x^2*tan(1/2*a*x)^4 - 64*a^3*x^3*tan(3/2*a*x) + 27*a^2*x^
2*real_part(cos_integral(3*a*x))*tan(3/2*a*x)^2 + a^2*x^2*real_part(cos_integral(a*x))*tan(3/2*a*x)^2 + a^2*x^
2*real_part(cos_integral(-a*x))*tan(3/2*a*x)^2 + 27*a^2*x^2*real_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2 + 1
2*a*x*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 28*a*x*tan(3/2*a*x)*tan(1/2*a*x)^4 + 16*a^2*x^2*tan(3/2*a*x)^2 - 56*a^2*
x^2*tan(3/2*a*x)*tan(1/2*a*x) + 24*a^2*x^2*tan(1/2*a*x)^2 + 8*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 27*a^2*x^2*real_
part(cos_integral(3*a*x)) + a^2*x^2*real_part(cos_integral(a*x)) + a^2*x^2*real_part(cos_integral(-a*x)) + 27*
a^2*x^2*real_part(cos_integral(-3*a*x)) - 20*a*x*tan(3/2*a*x)^2*tan(1/2*a*x) + 20*a*x*tan(1/2*a*x)^3 + 32*a^2*
x^2 - 12*tan(3/2*a*x)^2*tan(1/2*a*x)^2 + 4*tan(1/2*a*x)^4 - 28*a*x*tan(3/2*a*x) - 12*a*x*tan(1/2*a*x) + 4*tan(
3/2*a*x)^2 - 12*tan(1/2*a*x)^2 + 8)/(2*a^5*x^7*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - a^4*x^6*tan(3/2*a*x)^2*tan(1/2*
a*x)^4 + 2*a^5*x^7*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^5*x^7*tan(1/2*a*x)^3 - a^4*x^6*tan(1/2*a*x)^4 + 2*a^5*x^7
*tan(1/2*a*x) + 4*a^3*x^5*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + a^4*x^6*tan(3/2*a*x)^2 - 2*a^2*x^4*tan(3/2*a*x)^2*ta
n(1/2*a*x)^4 + 4*a^3*x^5*tan(3/2*a*x)^2*tan(1/2*a*x) + 4*a^3*x^5*tan(1/2*a*x)^3 + a^4*x^6 - 2*a^2*x^4*tan(1/2*
a*x)^4 + 4*a^3*x^5*tan(1/2*a*x) + 2*a*x^3*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 2*a^2*x^4*tan(3/2*a*x)^2 - x^2*tan(3
/2*a*x)^2*tan(1/2*a*x)^4 + 2*a*x^3*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a*x^3*tan(1/2*a*x)^3 + 2*a^2*x^4 - x^2*tan(
1/2*a*x)^4 + 2*a*x^3*tan(1/2*a*x) + x^2*tan(3/2*a*x)^2 + x^2)