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

Optimal. Leaf size=131 \[ -\frac{1}{8} a^2 \text{Si}(a x)+\frac{27}{8} a^2 \text{Si}(3 a x)+\frac{\sin ^3(a x)}{a^2 x^4}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}-\frac{3 \sin ^3(a x)}{2 x^2}+\frac{\sin (a x)}{x^2}+\frac{\sin ^2(a x) \cos (a x)}{a x^3}+\frac{a \cos (a x)}{x}-\frac{9 a \sin ^2(a x) \cos (a x)}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.227389, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4598, 3314, 3297, 3299, 3312} \[ -\frac{1}{8} a^2 \text{Si}(a x)+\frac{27}{8} a^2 \text{Si}(3 a x)+\frac{\sin ^3(a x)}{a^2 x^4}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}-\frac{3 \sin ^3(a x)}{2 x^2}+\frac{\sin (a x)}{x^2}+\frac{\sin ^2(a x) \cos (a x)}{a x^3}+\frac{a \cos (a x)}{x}-\frac{9 a \sin ^2(a x) \cos (a x)}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4598

Int[(((b_.)*(x_))^(m_)*Sin[(a_.)*(x_)]^(n_))/(Cos[(a_.)*(x_)]*(d_.)*(x_) + (c_.)*Sin[(a_.)*(x_)])^2, x_Symbol]
:> Simp[(b*(b*x)^(m - 1)*Sin[a*x]^(n - 1))/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])), x] - Dist[(b^2*(n - 1))/d^2, In
t[(b*x)^(m - 2)*Sin[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 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 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{\sin ^5(a x)}{x^3 (a x \cos (a x)-\sin (a x))^2} \, dx &=\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}-\frac{4 \int \frac{\sin ^3(a x)}{x^5} \, dx}{a^2}\\ &=\frac{\cos (a x) \sin ^2(a x)}{a x^3}+\frac{\sin ^3(a x)}{a^2 x^4}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}-2 \int \frac{\sin (a x)}{x^3} \, dx+3 \int \frac{\sin ^3(a x)}{x^3} \, dx\\ &=\frac{\sin (a x)}{x^2}+\frac{\cos (a x) \sin ^2(a x)}{a x^3}-\frac{9 a \cos (a x) \sin ^2(a x)}{2 x}+\frac{\sin ^3(a x)}{a^2 x^4}-\frac{3 \sin ^3(a x)}{2 x^2}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}-a \int \frac{\cos (a x)}{x^2} \, dx+\left (9 a^2\right ) \int \frac{\sin (a x)}{x} \, dx-\frac{1}{2} \left (27 a^2\right ) \int \frac{\sin ^3(a x)}{x} \, dx\\ &=\frac{a \cos (a x)}{x}+\frac{\sin (a x)}{x^2}+\frac{\cos (a x) \sin ^2(a x)}{a x^3}-\frac{9 a \cos (a x) \sin ^2(a x)}{2 x}+\frac{\sin ^3(a x)}{a^2 x^4}-\frac{3 \sin ^3(a x)}{2 x^2}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}+9 a^2 \text{Si}(a x)+a^2 \int \frac{\sin (a x)}{x} \, dx-\frac{1}{2} \left (27 a^2\right ) \int \left (\frac{3 \sin (a x)}{4 x}-\frac{\sin (3 a x)}{4 x}\right ) \, dx\\ &=\frac{a \cos (a x)}{x}+\frac{\sin (a x)}{x^2}+\frac{\cos (a x) \sin ^2(a x)}{a x^3}-\frac{9 a \cos (a x) \sin ^2(a x)}{2 x}+\frac{\sin ^3(a x)}{a^2 x^4}-\frac{3 \sin ^3(a x)}{2 x^2}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}+10 a^2 \text{Si}(a x)+\frac{1}{8} \left (27 a^2\right ) \int \frac{\sin (3 a x)}{x} \, dx-\frac{1}{8} \left (81 a^2\right ) \int \frac{\sin (a x)}{x} \, dx\\ &=\frac{a \cos (a x)}{x}+\frac{\sin (a x)}{x^2}+\frac{\cos (a x) \sin ^2(a x)}{a x^3}-\frac{9 a \cos (a x) \sin ^2(a x)}{2 x}+\frac{\sin ^3(a x)}{a^2 x^4}-\frac{3 \sin ^3(a x)}{2 x^2}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}-\frac{1}{8} a^2 \text{Si}(a x)+\frac{27}{8} a^2 \text{Si}(3 a x)\\ \end{align*}

Mathematica [A]  time = 0.996785, size = 142, normalized size = 1.08 \[ \frac{-2 a^2 x^2 \text{Si}(a x) (a x \cos (a x)-\sin (a x))+54 a^2 x^2 \text{Si}(3 a x) (a x \cos (a x)-\sin (a x))-a^2 x^2+8 a^2 x^2 \cos (2 a x)+9 a^2 x^2 \cos (4 a x)+12 a x \sin (2 a x)-6 a x \sin (4 a x)-4 \cos (2 a x)+\cos (4 a x)+3}{16 x^2 (a x \cos (a x)-\sin (a x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a*x]^5/(x^3*(a*x*Cos[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] + 12*a*x*Sin[2*a*x] - 6
*a*x*Sin[4*a*x] - 2*a^2*x^2*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[a*x] + 54*a^2*x^2*(a*x*Cos[a*x] - Sin[a*x])*
SinIntegral[3*a*x])/(16*x^2*(a*x*Cos[a*x] - Sin[a*x]))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a*x)^5/x^3/(a*x*cos(a*x)-sin(a*x))^2,x)

[Out]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.49318, size = 385, normalized size = 2.94 \begin{align*} \frac{4 \,{\left (9 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right )^{4} - 4 \,{\left (7 \, a^{2} x^{2} + 2\right )} \cos \left (a x\right )^{2} +{\left (27 \, a^{3} x^{3} \operatorname{Si}\left (3 \, a x\right ) - a^{3} x^{3} \operatorname{Si}\left (a x\right )\right )} \cos \left (a x\right ) -{\left (24 \, a x \cos \left (a x\right )^{3} + 27 \, a^{2} x^{2} \operatorname{Si}\left (3 \, a x\right ) - a^{2} x^{2} \operatorname{Si}\left (a x\right ) - 24 \, a x \cos \left (a x\right )\right )} \sin \left (a x\right ) + 4}{8 \,{\left (a x^{3} \cos \left (a x\right ) - x^{2} \sin \left (a x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(9*a^2*x^2 + 1)*cos(a*x)^4 - 4*(7*a^2*x^2 + 2)*cos(a*x)^2 + (27*a^3*x^3*sin_integral(3*a*x) - a^3*x^3*s
in_integral(a*x))*cos(a*x) - (24*a*x*cos(a*x)^3 + 27*a^2*x^2*sin_integral(3*a*x) - a^2*x^2*sin_integral(a*x) -
24*a*x*cos(a*x))*sin(a*x) + 4)/(a*x^3*cos(a*x) - x^2*sin(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(sin(a*x)**5/x**3/(a*x*cos(a*x)-sin(a*x))**2,x)

[Out]

Timed out

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Giac [C]  time = 1.85612, size = 5636, normalized size = 43.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(27*a^7*x^7*imag_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - a^7*x^7*imag_part(cos_integral
(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + a^7*x^7*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 2
7*a^7*x^7*imag_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 54*a^7*x^7*sin_integral(3*a*x)*tan(3
/2*a*x)^2*tan(1/2*a*x)^4 - 2*a^7*x^7*sin_integral(a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 27*a^7*x^7*imag_part(co
s_integral(3*a*x))*tan(1/2*a*x)^4 - a^7*x^7*imag_part(cos_integral(a*x))*tan(1/2*a*x)^4 + a^7*x^7*imag_part(co
s_integral(-a*x))*tan(1/2*a*x)^4 - 27*a^7*x^7*imag_part(cos_integral(-3*a*x))*tan(1/2*a*x)^4 + 54*a^7*x^7*sin_
integral(3*a*x)*tan(1/2*a*x)^4 - 2*a^7*x^7*sin_integral(a*x)*tan(1/2*a*x)^4 + 54*a^6*x^6*imag_part(cos_integra
l(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 2*a^6*x^6*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3
+ 2*a^6*x^6*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 54*a^6*x^6*imag_part(cos_integral(-
3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 108*a^6*x^6*sin_integral(3*a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 4*a^6*
x^6*sin_integral(a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 16*a^6*x^6*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 27*a^7*x^7*im
ag_part(cos_integral(3*a*x))*tan(3/2*a*x)^2 + a^7*x^7*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2 - a^7*x^7*im
ag_part(cos_integral(-a*x))*tan(3/2*a*x)^2 + 27*a^7*x^7*imag_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2 - 54*a^
7*x^7*sin_integral(3*a*x)*tan(3/2*a*x)^2 + 2*a^7*x^7*sin_integral(a*x)*tan(3/2*a*x)^2 + 54*a^5*x^5*imag_part(c
os_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 2*a^5*x^5*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(
1/2*a*x)^4 + 2*a^5*x^5*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 54*a^5*x^5*imag_part(cos_
integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 108*a^5*x^5*sin_integral(3*a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^
4 - 4*a^5*x^5*sin_integral(a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 54*a^6*x^6*imag_part(cos_integral(3*a*x))*tan(
3/2*a*x)^2*tan(1/2*a*x) - 2*a^6*x^6*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^6*x^6*imag_
part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) - 54*a^6*x^6*imag_part(cos_integral(-3*a*x))*tan(3/2*a*x)
^2*tan(1/2*a*x) + 108*a^6*x^6*sin_integral(3*a*x)*tan(3/2*a*x)^2*tan(1/2*a*x) - 4*a^6*x^6*sin_integral(a*x)*ta
n(3/2*a*x)^2*tan(1/2*a*x) - 4*a^6*x^6*tan(3/2*a*x)^2*tan(1/2*a*x)^2 + 54*a^6*x^6*imag_part(cos_integral(3*a*x)
)*tan(1/2*a*x)^3 - 2*a^6*x^6*imag_part(cos_integral(a*x))*tan(1/2*a*x)^3 + 2*a^6*x^6*imag_part(cos_integral(-a
*x))*tan(1/2*a*x)^3 - 54*a^6*x^6*imag_part(cos_integral(-3*a*x))*tan(1/2*a*x)^3 + 108*a^6*x^6*sin_integral(3*a
*x)*tan(1/2*a*x)^3 - 4*a^6*x^6*sin_integral(a*x)*tan(1/2*a*x)^3 + 20*a^6*x^6*tan(1/2*a*x)^4 - 27*a^7*x^7*imag_
part(cos_integral(3*a*x)) + a^7*x^7*imag_part(cos_integral(a*x)) - a^7*x^7*imag_part(cos_integral(-a*x)) + 27*
a^7*x^7*imag_part(cos_integral(-3*a*x)) - 54*a^7*x^7*sin_integral(3*a*x) + 2*a^7*x^7*sin_integral(a*x) - 36*a^
5*x^5*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 54*a^5*x^5*imag_part(cos_integral(3*a*x))*tan(1/2*a*x)^4 - 2*a^5*x^5*ima
g_part(cos_integral(a*x))*tan(1/2*a*x)^4 + 2*a^5*x^5*imag_part(cos_integral(-a*x))*tan(1/2*a*x)^4 - 54*a^5*x^5
*imag_part(cos_integral(-3*a*x))*tan(1/2*a*x)^4 + 108*a^5*x^5*sin_integral(3*a*x)*tan(1/2*a*x)^4 - 4*a^5*x^5*s
in_integral(a*x)*tan(1/2*a*x)^4 + 12*a^5*x^5*tan(3/2*a*x)*tan(1/2*a*x)^4 + 20*a^6*x^6*tan(3/2*a*x)^2 + 54*a^6*
x^6*imag_part(cos_integral(3*a*x))*tan(1/2*a*x) - 2*a^6*x^6*imag_part(cos_integral(a*x))*tan(1/2*a*x) + 2*a^6*
x^6*imag_part(cos_integral(-a*x))*tan(1/2*a*x) - 54*a^6*x^6*imag_part(cos_integral(-3*a*x))*tan(1/2*a*x) + 108
*a^6*x^6*sin_integral(3*a*x)*tan(1/2*a*x) - 4*a^6*x^6*sin_integral(a*x)*tan(1/2*a*x) - 4*a^6*x^6*tan(1/2*a*x)^
2 + 108*a^4*x^4*imag_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 4*a^4*x^4*imag_part(cos_integra
l(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 4*a^4*x^4*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3
- 108*a^4*x^4*imag_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 216*a^4*x^4*sin_integral(3*a*x)*
tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 8*a^4*x^4*sin_integral(a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 32*a^4*x^4*tan(3/2
*a*x)^2*tan(1/2*a*x)^4 - 54*a^5*x^5*imag_part(cos_integral(3*a*x))*tan(3/2*a*x)^2 + 2*a^5*x^5*imag_part(cos_in
tegral(a*x))*tan(3/2*a*x)^2 - 2*a^5*x^5*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2 + 54*a^5*x^5*imag_part(co
s_integral(-3*a*x))*tan(3/2*a*x)^2 - 108*a^5*x^5*sin_integral(3*a*x)*tan(3/2*a*x)^2 + 4*a^5*x^5*sin_integral(a
*x)*tan(3/2*a*x)^2 - 36*a^5*x^5*tan(3/2*a*x)^2*tan(1/2*a*x) + 36*a^5*x^5*tan(1/2*a*x)^3 + 27*a^3*x^3*imag_part
(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - a^3*x^3*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(
1/2*a*x)^4 + a^3*x^3*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 27*a^3*x^3*imag_part(cos_in
tegral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 54*a^3*x^3*sin_integral(3*a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^4 -
2*a^3*x^3*sin_integral(a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 16*a^6*x^6 + 108*a^4*x^4*imag_part(cos_integral(3
*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) - 4*a^4*x^4*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 4*a^
4*x^4*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) - 108*a^4*x^4*imag_part(cos_integral(-3*a*x))*
tan(3/2*a*x)^2*tan(1/2*a*x) + 216*a^4*x^4*sin_integral(3*a*x)*tan(3/2*a*x)^2*tan(1/2*a*x) - 8*a^4*x^4*sin_inte
gral(a*x)*tan(3/2*a*x)^2*tan(1/2*a*x) - 8*a^4*x^4*tan(3/2*a*x)^2*tan(1/2*a*x)^2 + 108*a^4*x^4*imag_part(cos_in
tegral(3*a*x))*tan(1/2*a*x)^3 - 4*a^4*x^4*imag_part(cos_integral(a*x))*tan(1/2*a*x)^3 + 4*a^4*x^4*imag_part(co
s_integral(-a*x))*tan(1/2*a*x)^3 - 108*a^4*x^4*imag_part(cos_integral(-3*a*x))*tan(1/2*a*x)^3 + 216*a^4*x^4*si
n_integral(3*a*x)*tan(1/2*a*x)^3 - 8*a^4*x^4*sin_integral(a*x)*tan(1/2*a*x)^3 + 24*a^4*x^4*tan(3/2*a*x)*tan(1/
2*a*x)^3 + 32*a^4*x^4*tan(1/2*a*x)^4 - 54*a^5*x^5*imag_part(cos_integral(3*a*x)) + 2*a^5*x^5*imag_part(cos_int
egral(a*x)) - 2*a^5*x^5*imag_part(cos_integral(-a*x)) + 54*a^5*x^5*imag_part(cos_integral(-3*a*x)) - 108*a^5*x
^5*sin_integral(3*a*x) + 4*a^5*x^5*sin_integral(a*x) - 12*a^5*x^5*tan(3/2*a*x) + 36*a^5*x^5*tan(1/2*a*x) - 48*
a^3*x^3*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 27*a^3*x^3*imag_part(cos_integral(3*a*x))*tan(1/2*a*x)^4 - a^3*x^3*ima
g_part(cos_integral(a*x))*tan(1/2*a*x)^4 + a^3*x^3*imag_part(cos_integral(-a*x))*tan(1/2*a*x)^4 - 27*a^3*x^3*i
mag_part(cos_integral(-3*a*x))*tan(1/2*a*x)^4 + 54*a^3*x^3*sin_integral(3*a*x)*tan(1/2*a*x)^4 - 2*a^3*x^3*sin_
integral(a*x)*tan(1/2*a*x)^4 + 16*a^3*x^3*tan(3/2*a*x)*tan(1/2*a*x)^4 + 32*a^4*x^4*tan(3/2*a*x)^2 + 108*a^4*x^
4*imag_part(cos_integral(3*a*x))*tan(1/2*a*x) - 4*a^4*x^4*imag_part(cos_integral(a*x))*tan(1/2*a*x) + 4*a^4*x^
4*imag_part(cos_integral(-a*x))*tan(1/2*a*x) - 108*a^4*x^4*imag_part(cos_integral(-3*a*x))*tan(1/2*a*x) + 216*
a^4*x^4*sin_integral(3*a*x)*tan(1/2*a*x) - 8*a^4*x^4*sin_integral(a*x)*tan(1/2*a*x) + 24*a^4*x^4*tan(3/2*a*x)*
tan(1/2*a*x) - 8*a^4*x^4*tan(1/2*a*x)^2 + 54*a^2*x^2*imag_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x
)^3 - 2*a^2*x^2*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 2*a^2*x^2*imag_part(cos_integral(
-a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 54*a^2*x^2*imag_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x)^
3 + 108*a^2*x^2*sin_integral(3*a*x)*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - 4*a^2*x^2*sin_integral(a*x)*tan(3/2*a*x)^2
*tan(1/2*a*x)^3 - 32*a^2*x^2*tan(3/2*a*x)^2*tan(1/2*a*x)^4 - 27*a^3*x^3*imag_part(cos_integral(3*a*x))*tan(3/2
*a*x)^2 + a^3*x^3*imag_part(cos_integral(a*x))*tan(3/2*a*x)^2 - a^3*x^3*imag_part(cos_integral(-a*x))*tan(3/2*
a*x)^2 + 27*a^3*x^3*imag_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2 - 54*a^3*x^3*sin_integral(3*a*x)*tan(3/2*a*
x)^2 + 2*a^3*x^3*sin_integral(a*x)*tan(3/2*a*x)^2 - 80*a^3*x^3*tan(3/2*a*x)^2*tan(1/2*a*x) + 80*a^3*x^3*tan(1/
2*a*x)^3 - 32*a^4*x^4 + 54*a^2*x^2*imag_part(cos_integral(3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) - 2*a^2*x^2*imag
_part(cos_integral(a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^2*x^2*imag_part(cos_integral(-a*x))*tan(3/2*a*x)^2*
tan(1/2*a*x) - 54*a^2*x^2*imag_part(cos_integral(-3*a*x))*tan(3/2*a*x)^2*tan(1/2*a*x) + 108*a^2*x^2*sin_integr
al(3*a*x)*tan(3/2*a*x)^2*tan(1/2*a*x) - 4*a^2*x^2*sin_integral(a*x)*tan(3/2*a*x)^2*tan(1/2*a*x) - 60*a^2*x^2*t
an(3/2*a*x)^2*tan(1/2*a*x)^2 + 54*a^2*x^2*imag_part(cos_integral(3*a*x))*tan(1/2*a*x)^3 - 2*a^2*x^2*imag_part(
cos_integral(a*x))*tan(1/2*a*x)^3 + 2*a^2*x^2*imag_part(cos_integral(-a*x))*tan(1/2*a*x)^3 - 54*a^2*x^2*imag_p
art(cos_integral(-3*a*x))*tan(1/2*a*x)^3 + 108*a^2*x^2*sin_integral(3*a*x)*tan(1/2*a*x)^3 - 4*a^2*x^2*sin_inte
gral(a*x)*tan(1/2*a*x)^3 + 32*a^2*x^2*tan(3/2*a*x)*tan(1/2*a*x)^3 - 4*a^2*x^2*tan(1/2*a*x)^4 - 27*a^3*x^3*imag
_part(cos_integral(3*a*x)) + a^3*x^3*imag_part(cos_integral(a*x)) - a^3*x^3*imag_part(cos_integral(-a*x)) + 27
*a^3*x^3*imag_part(cos_integral(-3*a*x)) - 54*a^3*x^3*sin_integral(3*a*x) + 2*a^3*x^3*sin_integral(a*x) - 16*a
^3*x^3*tan(3/2*a*x) + 48*a^3*x^3*tan(1/2*a*x) - 12*a*x*tan(3/2*a*x)^2*tan(1/2*a*x)^3 + 4*a*x*tan(3/2*a*x)*tan(
1/2*a*x)^4 - 4*a^2*x^2*tan(3/2*a*x)^2 + 54*a^2*x^2*imag_part(cos_integral(3*a*x))*tan(1/2*a*x) - 2*a^2*x^2*ima
g_part(cos_integral(a*x))*tan(1/2*a*x) + 2*a^2*x^2*imag_part(cos_integral(-a*x))*tan(1/2*a*x) - 54*a^2*x^2*ima
g_part(cos_integral(-3*a*x))*tan(1/2*a*x) + 108*a^2*x^2*sin_integral(3*a*x)*tan(1/2*a*x) - 4*a^2*x^2*sin_integ
ral(a*x)*tan(1/2*a*x) + 32*a^2*x^2*tan(3/2*a*x)*tan(1/2*a*x) - 60*a^2*x^2*tan(1/2*a*x)^2 - 44*a*x*tan(3/2*a*x)
^2*tan(1/2*a*x) + 44*a*x*tan(1/2*a*x)^3 - 32*a^2*x^2 - 24*tan(3/2*a*x)^2*tan(1/2*a*x)^2 + 8*tan(3/2*a*x)*tan(1
/2*a*x)^3 - 4*a*x*tan(3/2*a*x) + 12*a*x*tan(1/2*a*x) + 8*tan(3/2*a*x)*tan(1/2*a*x) - 24*tan(1/2*a*x)^2)/(a^5*x
^7*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + a^5*x^7*tan(1/2*a*x)^4 + 2*a^4*x^6*tan(3/2*a*x)^2*tan(1/2*a*x)^3 - a^5*x^7*
tan(3/2*a*x)^2 + 2*a^3*x^5*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 2*a^4*x^6*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*a^4*x^6*t
an(1/2*a*x)^3 - a^5*x^7 + 2*a^3*x^5*tan(1/2*a*x)^4 + 2*a^4*x^6*tan(1/2*a*x) + 4*a^2*x^4*tan(3/2*a*x)^2*tan(1/2
*a*x)^3 - 2*a^3*x^5*tan(3/2*a*x)^2 + a*x^3*tan(3/2*a*x)^2*tan(1/2*a*x)^4 + 4*a^2*x^4*tan(3/2*a*x)^2*tan(1/2*a*
x) + 4*a^2*x^4*tan(1/2*a*x)^3 - 2*a^3*x^5 + a*x^3*tan(1/2*a*x)^4 + 4*a^2*x^4*tan(1/2*a*x) + 2*x^2*tan(3/2*a*x)
^2*tan(1/2*a*x)^3 - a*x^3*tan(3/2*a*x)^2 + 2*x^2*tan(3/2*a*x)^2*tan(1/2*a*x) + 2*x^2*tan(1/2*a*x)^3 - a*x^3 +
2*x^2*tan(1/2*a*x))