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

Optimal. Leaf size=175 \[ -\frac {2}{3} a^3 \text {Si}(2 a x)+\frac {16}{3} a^3 \text {Si}(4 a x)+\frac {\sin ^4(a x)}{a^2 x^5}+\frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}+\frac {a^2}{x}+\frac {32 a^2 \sin ^4(a x)}{3 x}-\frac {10 a^2 \sin ^2(a x)}{x}+\frac {\sin ^3(a x) \cos (a x)}{a x^4}-\frac {4 \sin ^4(a x)}{3 x^3}+\frac {\sin ^2(a x)}{x^3}-\frac {8 a \sin ^3(a x) \cos (a x)}{3 x^2}+\frac {a \sin (a x) \cos (a x)}{x^2} \]

[Out]

a^2/x-2/3*a^3*Si(2*a*x)+16/3*a^3*Si(4*a*x)+a*cos(a*x)*sin(a*x)/x^2+sin(a*x)^2/x^3-10*a^2*sin(a*x)^2/x+cos(a*x)
*sin(a*x)^3/a/x^4-8/3*a*cos(a*x)*sin(a*x)^3/x^2+sin(a*x)^4/a^2/x^5-4/3*sin(a*x)^4/x^3+32/3*a^2*sin(a*x)^4/x+si
n(a*x)^5/a^2/x^5/(a*x*cos(a*x)-sin(a*x))

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Rubi [A]  time = 0.30, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4598, 3314, 30, 3313, 12, 3299} \[ -\frac {2}{3} a^3 \text {Si}(2 a x)+\frac {16}{3} a^3 \text {Si}(4 a x)+\frac {\sin ^4(a x)}{a^2 x^5}+\frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}+\frac {a^2}{x}+\frac {32 a^2 \sin ^4(a x)}{3 x}-\frac {10 a^2 \sin ^2(a x)}{x}-\frac {4 \sin ^4(a x)}{3 x^3}+\frac {\sin ^2(a x)}{x^3}-\frac {8 a \sin ^3(a x) \cos (a x)}{3 x^2}+\frac {\sin ^3(a x) \cos (a x)}{a x^4}+\frac {a \sin (a x) \cos (a x)}{x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 30

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

Rubi steps

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

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Mathematica [A]  time = 1.45, size = 198, normalized size = 1.13 \[ \frac {-32 a^3 x^3 \text {Si}(2 a x) (a x \cos (a x)-\sin (a x))+256 a^3 x^3 \text {Si}(4 a x) (a x \cos (a x)-\sin (a x))-8 a^3 x^3 \cos (a x)+24 a^3 x^3 \cos (3 a x)+32 a^3 x^3 \cos (5 a x)-12 a^2 x^2 \sin (a x)+44 a^2 x^2 \sin (3 a x)-24 a^2 x^2 \sin (5 a x)+10 \sin (a x)-5 \sin (3 a x)+\sin (5 a x)+8 a x \cos (a x)-12 a x \cos (3 a x)+4 a x \cos (5 a x)}{48 x^3 (a x \cos (a x)-\sin (a x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.07, size = 186, normalized size = 1.06 \[ \frac {4 \, {\left (8 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{5} - 2 \, {\left (17 \, a^{3} x^{3} + 4 \, a x\right )} \cos \left (a x\right )^{3} + {\left (16 \, a^{4} x^{4} \operatorname {Si}\left (4 \, a x\right ) - 2 \, a^{4} x^{4} \operatorname {Si}\left (2 \, a x\right ) + 5 \, a^{3} x^{3} + 4 \, a x\right )} \cos \left (a x\right ) - {\left (16 \, a^{3} x^{3} \operatorname {Si}\left (4 \, a x\right ) - 2 \, a^{3} x^{3} \operatorname {Si}\left (2 \, a x\right ) + {\left (24 \, a^{2} x^{2} - 1\right )} \cos \left (a x\right )^{4} + 5 \, a^{2} x^{2} - {\left (29 \, a^{2} x^{2} - 2\right )} \cos \left (a x\right )^{2} - 1\right )} \sin \left (a x\right )}{3 \, {\left (a x^{4} \cos \left (a x\right ) - x^{3} \sin \left (a x\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(4*(8*a^3*x^3 + a*x)*cos(a*x)^5 - 2*(17*a^3*x^3 + 4*a*x)*cos(a*x)^3 + (16*a^4*x^4*sin_integral(4*a*x) - 2*
a^4*x^4*sin_integral(2*a*x) + 5*a^3*x^3 + 4*a*x)*cos(a*x) - (16*a^3*x^3*sin_integral(4*a*x) - 2*a^3*x^3*sin_in
tegral(2*a*x) + (24*a^2*x^2 - 1)*cos(a*x)^4 + 5*a^2*x^2 - (29*a^2*x^2 - 2)*cos(a*x)^2 - 1)*sin(a*x))/(a*x^4*co
s(a*x) - x^3*sin(a*x))

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giac [C]  time = 1.20, size = 7347, normalized size = 41.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(32*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)^2 - 4*a^8*x^8*imag_part(c
os_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 4*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)^2 - 32*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)^2 + 64*a^8*x^8*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^8*x^8*sin_integral(2*a*x)*
tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 32*a^8*x^8*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 4
*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 4*a^8*x^8*imag_part(cos_integral(-2*a*x))*ta
n(2*a*x)^2*tan(a*x)^2 + 32*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 64*a^8*x^8*sin_in
tegral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 8*a^8*x^8*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 32*a^8*x^8*ima
g_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 4*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(2*a*x)^
2*tan(1/2*a*x)^2 + 4*a^8*x^8*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 32*a^8*x^8*imag_par
t(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 64*a^8*x^8*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x)
^2 - 8*a^8*x^8*sin_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 + 32*a^8*x^8*imag_part(cos_integral(4*a*x))*tan
(a*x)^2*tan(1/2*a*x)^2 - 4*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^8*x^8*imag_p
art(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 32*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*ta
n(1/2*a*x)^2 + 64*a^8*x^8*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^8*x^8*sin_integral(2*a*x)*tan(a*
x)^2*tan(1/2*a*x)^2 + 64*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) - 8*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) + 8*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) - 64*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) + 128*a^7*x^7*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 16*a^7*x^7*sin_integral
(2*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 12*a^7*x^7*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 32*a^8*x^8*
imag_part(cos_integral(4*a*x))*tan(2*a*x)^2 + 4*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2 - 4*a^8*x^
8*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2 + 32*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2 - 64*
a^8*x^8*sin_integral(4*a*x)*tan(2*a*x)^2 + 8*a^8*x^8*sin_integral(2*a*x)*tan(2*a*x)^2 - 32*a^8*x^8*imag_part(c
os_integral(4*a*x))*tan(a*x)^2 + 4*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(a*x)^2 - 4*a^8*x^8*imag_part(cos
_integral(-2*a*x))*tan(a*x)^2 + 32*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(a*x)^2 - 64*a^8*x^8*sin_integra
l(4*a*x)*tan(a*x)^2 + 8*a^8*x^8*sin_integral(2*a*x)*tan(a*x)^2 + 32*a^8*x^8*imag_part(cos_integral(4*a*x))*tan
(1/2*a*x)^2 - 4*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(1/2*a*x)^2 + 4*a^8*x^8*imag_part(cos_integral(-2*a*
x))*tan(1/2*a*x)^2 - 32*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(1/2*a*x)^2 + 64*a^8*x^8*sin_integral(4*a*x
)*tan(1/2*a*x)^2 - 8*a^8*x^8*sin_integral(2*a*x)*tan(1/2*a*x)^2 + 64*a^6*x^6*imag_part(cos_integral(4*a*x))*ta
n(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 8*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)^2 + 8*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)^2 - 64*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)^2 + 128*a^6*x^6*sin_integral(4*a*x)*tan(2*a*x)
^2*tan(a*x)^2*tan(1/2*a*x)^2 - 16*a^6*x^6*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 + 12*a^7*
x^7*tan(2*a*x)^2*tan(a*x)^2 + 64*a^7*x^7*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 8*a^7*x^7*
imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 8*a^7*x^7*imag_part(cos_integral(-2*a*x))*tan(2*a*x
)^2*tan(1/2*a*x) - 64*a^7*x^7*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 128*a^7*x^7*sin_inte
gral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x) - 16*a^7*x^7*sin_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x) + 64*a^7*x^7*
imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*x) - 8*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(a*x)^2*t
an(1/2*a*x) + 8*a^7*x^7*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 64*a^7*x^7*imag_part(cos_int
egral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x) + 128*a^7*x^7*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x) - 16*a^7*x^7*
sin_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x) - 20*a^7*x^7*tan(2*a*x)^2*tan(1/2*a*x)^2 + 20*a^7*x^7*tan(a*x)^2*t
an(1/2*a*x)^2 - 32*a^8*x^8*imag_part(cos_integral(4*a*x)) + 4*a^8*x^8*imag_part(cos_integral(2*a*x)) - 4*a^8*x
^8*imag_part(cos_integral(-2*a*x)) + 32*a^8*x^8*imag_part(cos_integral(-4*a*x)) - 64*a^8*x^8*sin_integral(4*a*
x) + 8*a^8*x^8*sin_integral(2*a*x) - 64*a^6*x^6*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 8*a^6
*x^6*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 8*a^6*x^6*imag_part(cos_integral(-2*a*x))*tan(2*
a*x)^2*tan(a*x)^2 + 64*a^6*x^6*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 128*a^6*x^6*sin_integ
ral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 16*a^6*x^6*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2 - 24*a^6*x^6*tan(2
*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 64*a^6*x^6*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 8*a^
6*x^6*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 8*a^6*x^6*imag_part(cos_integral(-2*a*x))*t
an(2*a*x)^2*tan(1/2*a*x)^2 - 64*a^6*x^6*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 128*a^6*
x^6*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 - 16*a^6*x^6*sin_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x)
^2 - 4*a^6*x^6*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x)^2 + 64*a^6*x^6*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan
(1/2*a*x)^2 - 8*a^6*x^6*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 8*a^6*x^6*imag_part(cos_int
egral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 64*a^6*x^6*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x)^
2 + 128*a^6*x^6*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 16*a^6*x^6*sin_integral(2*a*x)*tan(a*x)^2*tan(
1/2*a*x)^2 + 8*a^6*x^6*tan(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 + 20*a^7*x^7*tan(2*a*x)^2 - 20*a^7*x^7*tan(a*x)^2
+ 64*a^7*x^7*imag_part(cos_integral(4*a*x))*tan(1/2*a*x) - 8*a^7*x^7*imag_part(cos_integral(2*a*x))*tan(1/2*a*
x) + 8*a^7*x^7*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x) - 64*a^7*x^7*imag_part(cos_integral(-4*a*x))*tan(1
/2*a*x) + 128*a^7*x^7*sin_integral(4*a*x)*tan(1/2*a*x) - 16*a^7*x^7*sin_integral(2*a*x)*tan(1/2*a*x) + 128*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) - 16*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) + 16*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) - 128*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) + 256*a^5*x
^5*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 32*a^5*x^5*sin_integral(2*a*x)*tan(2*a*x)^2*tan(
a*x)^2*tan(1/2*a*x) + 12*a^7*x^7*tan(1/2*a*x)^2 - 24*a^5*x^5*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 64*a^6*x
^6*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2 + 8*a^6*x^6*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2 - 8*a^6
*x^6*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2 + 64*a^6*x^6*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2 -
128*a^6*x^6*sin_integral(4*a*x)*tan(2*a*x)^2 + 16*a^6*x^6*sin_integral(2*a*x)*tan(2*a*x)^2 + 4*a^6*x^6*tan(2*a
*x)^2*tan(a*x) - 64*a^6*x^6*imag_part(cos_integral(4*a*x))*tan(a*x)^2 + 8*a^6*x^6*imag_part(cos_integral(2*a*x
))*tan(a*x)^2 - 8*a^6*x^6*imag_part(cos_integral(-2*a*x))*tan(a*x)^2 + 64*a^6*x^6*imag_part(cos_integral(-4*a*
x))*tan(a*x)^2 - 128*a^6*x^6*sin_integral(4*a*x)*tan(a*x)^2 + 16*a^6*x^6*sin_integral(2*a*x)*tan(a*x)^2 - 8*a^
6*x^6*tan(2*a*x)*tan(a*x)^2 - 40*a^6*x^6*tan(2*a*x)^2*tan(1/2*a*x) + 40*a^6*x^6*tan(a*x)^2*tan(1/2*a*x) + 64*a
^6*x^6*imag_part(cos_integral(4*a*x))*tan(1/2*a*x)^2 - 8*a^6*x^6*imag_part(cos_integral(2*a*x))*tan(1/2*a*x)^2
+ 8*a^6*x^6*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x)^2 - 64*a^6*x^6*imag_part(cos_integral(-4*a*x))*tan(1
/2*a*x)^2 + 128*a^6*x^6*sin_integral(4*a*x)*tan(1/2*a*x)^2 - 16*a^6*x^6*sin_integral(2*a*x)*tan(1/2*a*x)^2 + 8
*a^6*x^6*tan(2*a*x)*tan(1/2*a*x)^2 - 4*a^6*x^6*tan(a*x)*tan(1/2*a*x)^2 + 32*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)^2 - 4*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)^2 + 4*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)^2 - 32*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)^2 + 64*a^4*x^4*sin_integral(4*a*x)*tan(
2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^4*x^4*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 12
*a^7*x^7 + 24*a^5*x^5*tan(2*a*x)^2*tan(a*x)^2 + 128*a^5*x^5*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/
2*a*x) - 16*a^5*x^5*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 16*a^5*x^5*imag_part(cos_integr
al(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 128*a^5*x^5*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)
+ 256*a^5*x^5*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x) - 32*a^5*x^5*sin_integral(2*a*x)*tan(2*a*x)^2*tan(
1/2*a*x) - 8*a^5*x^5*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x) + 128*a^5*x^5*imag_part(cos_integral(4*a*x))*tan(a*x)^
2*tan(1/2*a*x) - 16*a^5*x^5*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x) + 16*a^5*x^5*imag_part(cos_
integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 128*a^5*x^5*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x
) + 256*a^5*x^5*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x) - 32*a^5*x^5*sin_integral(2*a*x)*tan(a*x)^2*tan(1/
2*a*x) + 16*a^5*x^5*tan(2*a*x)*tan(a*x)^2*tan(1/2*a*x) - 36*a^5*x^5*tan(2*a*x)^2*tan(1/2*a*x)^2 + 36*a^5*x^5*t
an(a*x)^2*tan(1/2*a*x)^2 - 64*a^6*x^6*imag_part(cos_integral(4*a*x)) + 8*a^6*x^6*imag_part(cos_integral(2*a*x)
) - 8*a^6*x^6*imag_part(cos_integral(-2*a*x)) + 64*a^6*x^6*imag_part(cos_integral(-4*a*x)) - 128*a^6*x^6*sin_i
ntegral(4*a*x) + 16*a^6*x^6*sin_integral(2*a*x) - 8*a^6*x^6*tan(2*a*x) + 4*a^6*x^6*tan(a*x) - 32*a^4*x^4*imag_
part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 4*a^4*x^4*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(
a*x)^2 - 4*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(a*x)^2 + 32*a^4*x^4*imag_part(cos_integral
(-4*a*x))*tan(2*a*x)^2*tan(a*x)^2 - 64*a^4*x^4*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 8*a^4*x^4*sin_int
egral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 24*a^6*x^6*tan(1/2*a*x) - 48*a^4*x^4*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*
x) + 32*a^4*x^4*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 4*a^4*x^4*imag_part(cos_integral(
2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 4*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 -
32*a^4*x^4*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 64*a^4*x^4*sin_integral(4*a*x)*tan(2*
a*x)^2*tan(1/2*a*x)^2 - 8*a^4*x^4*sin_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*x)^2 - 2*a^4*x^4*tan(2*a*x)^2*tan
(a*x)*tan(1/2*a*x)^2 + 32*a^4*x^4*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 4*a^4*x^4*imag_pa
rt(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 4*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1
/2*a*x)^2 - 32*a^4*x^4*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + 64*a^4*x^4*sin_integral(4*a
*x)*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^4*x^4*sin_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 + 13*a^4*x^4*tan(2*a*x
)*tan(a*x)^2*tan(1/2*a*x)^2 + 36*a^5*x^5*tan(2*a*x)^2 - 36*a^5*x^5*tan(a*x)^2 + 128*a^5*x^5*imag_part(cos_inte
gral(4*a*x))*tan(1/2*a*x) - 16*a^5*x^5*imag_part(cos_integral(2*a*x))*tan(1/2*a*x) + 16*a^5*x^5*imag_part(cos_
integral(-2*a*x))*tan(1/2*a*x) - 128*a^5*x^5*imag_part(cos_integral(-4*a*x))*tan(1/2*a*x) + 256*a^5*x^5*sin_in
tegral(4*a*x)*tan(1/2*a*x) - 32*a^5*x^5*sin_integral(2*a*x)*tan(1/2*a*x) + 16*a^5*x^5*tan(2*a*x)*tan(1/2*a*x)
- 8*a^5*x^5*tan(a*x)*tan(1/2*a*x) + 64*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) - 8*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) + 8*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) - 64*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) + 128*a^3*x^3*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 16*a^3*x^
3*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) + 24*a^5*x^5*tan(1/2*a*x)^2 - 3*a^3*x^3*tan(2*a*x)^
2*tan(a*x)^2*tan(1/2*a*x)^2 - 32*a^4*x^4*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2 + 4*a^4*x^4*imag_part(cos
_integral(2*a*x))*tan(2*a*x)^2 - 4*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(2*a*x)^2 + 32*a^4*x^4*imag_part
(cos_integral(-4*a*x))*tan(2*a*x)^2 - 64*a^4*x^4*sin_integral(4*a*x)*tan(2*a*x)^2 + 8*a^4*x^4*sin_integral(2*a
*x)*tan(2*a*x)^2 + 2*a^4*x^4*tan(2*a*x)^2*tan(a*x) - 32*a^4*x^4*imag_part(cos_integral(4*a*x))*tan(a*x)^2 + 4*
a^4*x^4*imag_part(cos_integral(2*a*x))*tan(a*x)^2 - 4*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(a*x)^2 + 32*
a^4*x^4*imag_part(cos_integral(-4*a*x))*tan(a*x)^2 - 64*a^4*x^4*sin_integral(4*a*x)*tan(a*x)^2 + 8*a^4*x^4*sin
_integral(2*a*x)*tan(a*x)^2 - 13*a^4*x^4*tan(2*a*x)*tan(a*x)^2 - 72*a^4*x^4*tan(2*a*x)^2*tan(1/2*a*x) + 72*a^4
*x^4*tan(a*x)^2*tan(1/2*a*x) + 32*a^4*x^4*imag_part(cos_integral(4*a*x))*tan(1/2*a*x)^2 - 4*a^4*x^4*imag_part(
cos_integral(2*a*x))*tan(1/2*a*x)^2 + 4*a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x)^2 - 32*a^4*x^4*im
ag_part(cos_integral(-4*a*x))*tan(1/2*a*x)^2 + 64*a^4*x^4*sin_integral(4*a*x)*tan(1/2*a*x)^2 - 8*a^4*x^4*sin_i
ntegral(2*a*x)*tan(1/2*a*x)^2 + 13*a^4*x^4*tan(2*a*x)*tan(1/2*a*x)^2 - 2*a^4*x^4*tan(a*x)*tan(1/2*a*x)^2 - 24*
a^5*x^5 + 3*a^3*x^3*tan(2*a*x)^2*tan(a*x)^2 + 64*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a
*x) - 8*a^3*x^3*imag_part(cos_integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 8*a^3*x^3*imag_part(cos_integral(-2
*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 64*a^3*x^3*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x) + 128*
a^3*x^3*sin_integral(4*a*x)*tan(2*a*x)^2*tan(1/2*a*x) - 16*a^3*x^3*sin_integral(2*a*x)*tan(2*a*x)^2*tan(1/2*a*
x) - 4*a^3*x^3*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x) + 64*a^3*x^3*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan(1
/2*a*x) - 8*a^3*x^3*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x) + 8*a^3*x^3*imag_part(cos_integral(
-2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 64*a^3*x^3*imag_part(cos_integral(-4*a*x))*tan(a*x)^2*tan(1/2*a*x) + 128*a^
3*x^3*sin_integral(4*a*x)*tan(a*x)^2*tan(1/2*a*x) - 16*a^3*x^3*sin_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x) + 2
6*a^3*x^3*tan(2*a*x)*tan(a*x)^2*tan(1/2*a*x) - 15*a^3*x^3*tan(2*a*x)^2*tan(1/2*a*x)^2 + 24*a^3*x^3*tan(a*x)^2*
tan(1/2*a*x)^2 - 32*a^4*x^4*imag_part(cos_integral(4*a*x)) + 4*a^4*x^4*imag_part(cos_integral(2*a*x)) - 4*a^4*
x^4*imag_part(cos_integral(-2*a*x)) + 32*a^4*x^4*imag_part(cos_integral(-4*a*x)) - 64*a^4*x^4*sin_integral(4*a
*x) + 8*a^4*x^4*sin_integral(2*a*x) - 13*a^4*x^4*tan(2*a*x) + 2*a^4*x^4*tan(a*x) + 48*a^4*x^4*tan(1/2*a*x) - 3
0*a^2*x^2*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 10*a^2*x^2*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x)^2 + 5*a^2*x^2*t
an(2*a*x)*tan(a*x)^2*tan(1/2*a*x)^2 + 15*a^3*x^3*tan(2*a*x)^2 - 24*a^3*x^3*tan(a*x)^2 + 64*a^3*x^3*imag_part(c
os_integral(4*a*x))*tan(1/2*a*x) - 8*a^3*x^3*imag_part(cos_integral(2*a*x))*tan(1/2*a*x) + 8*a^3*x^3*imag_part
(cos_integral(-2*a*x))*tan(1/2*a*x) - 64*a^3*x^3*imag_part(cos_integral(-4*a*x))*tan(1/2*a*x) + 128*a^3*x^3*si
n_integral(4*a*x)*tan(1/2*a*x) - 16*a^3*x^3*sin_integral(2*a*x)*tan(1/2*a*x) + 26*a^3*x^3*tan(2*a*x)*tan(1/2*a
*x) - 4*a^3*x^3*tan(a*x)*tan(1/2*a*x) + 12*a^3*x^3*tan(1/2*a*x)^2 - 3*a*x*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)
^2 + 10*a^2*x^2*tan(2*a*x)^2*tan(a*x) - 5*a^2*x^2*tan(2*a*x)*tan(a*x)^2 - 54*a^2*x^2*tan(2*a*x)^2*tan(1/2*a*x)
+ 24*a^2*x^2*tan(a*x)^2*tan(1/2*a*x) + 5*a^2*x^2*tan(2*a*x)*tan(1/2*a*x)^2 - 10*a^2*x^2*tan(a*x)*tan(1/2*a*x)
^2 - 12*a^3*x^3 + 3*a*x*tan(2*a*x)^2*tan(a*x)^2 - 20*a*x*tan(2*a*x)^2*tan(a*x)*tan(1/2*a*x) + 10*a*x*tan(2*a*x
)*tan(a*x)^2*tan(1/2*a*x) + a*x*tan(2*a*x)^2*tan(1/2*a*x)^2 - 4*a*x*tan(a*x)^2*tan(1/2*a*x)^2 - 5*a^2*x^2*tan(
2*a*x) + 10*a^2*x^2*tan(a*x) - 6*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - a*x*tan(2*a*x)^2 + 4*a*x*tan(a*x)^2 +
10*a*x*tan(2*a*x)*tan(1/2*a*x) - 20*a*x*tan(a*x)*tan(1/2*a*x) + 2*tan(2*a*x)^2*tan(1/2*a*x) - 8*tan(a*x)^2*tan
(1/2*a*x))/(a^5*x^8*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - a^5*x^8*tan(2*a*x)^2*tan(a*x)^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)^2 + 2*a^4*x^7*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - a^
5*x^8*tan(2*a*x)^2 - a^5*x^8*tan(a*x)^2 + a^5*x^8*tan(1/2*a*x)^2 + 2*a^3*x^6*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a
*x)^2 + 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) - a^5*x^8 - 2*a^3*x^6*tan(2*a*
x)^2*tan(a*x)^2 + 2*a^3*x^6*tan(2*a*x)^2*tan(1/2*a*x)^2 + 2*a^3*x^6*tan(a*x)^2*tan(1/2*a*x)^2 + 2*a^4*x^7*tan(
1/2*a*x) + 4*a^2*x^5*tan(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x) - 2*a^3*x^6*tan(2*a*x)^2 - 2*a^3*x^6*tan(a*x)^2 + 2*
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 + 4*a^2*x^5*tan(2*a*x)^2*tan(1/2*a*x) +
4*a^2*x^5*tan(a*x)^2*tan(1/2*a*x) - 2*a^3*x^6 - a*x^4*tan(2*a*x)^2*tan(a*x)^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 + 4*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) -
a*x^4*tan(2*a*x)^2 - a*x^4*tan(a*x)^2 + a*x^4*tan(1/2*a*x)^2 + 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) - a*x^4 + 2*x^3*tan(1/2*a*x))

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{6}\left (a x \right )}{x^{4} \left (a x \cos \left (a x \right )-\sin \left (a x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a\,x\right )}^6}{x^4\,{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{6}{\left (a x \right )}}{x^{4} \left (a x \cos {\left (a x \right )} - \sin {\left (a x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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