Integrand size = 24, antiderivative size = 176 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a^2}{x}+\frac {\cos ^2(a x)}{x^3}-\frac {10 a^2 \cos ^2(a x)}{x}+\frac {\cos ^4(a x)}{a^2 x^5}-\frac {4 \cos ^4(a x)}{3 x^3}+\frac {32 a^2 \cos ^4(a x)}{3 x}-\frac {a \cos (a x) \sin (a x)}{x^2}-\frac {\cos ^3(a x) \sin (a x)}{a x^4}+\frac {8 a \cos ^3(a x) \sin (a x)}{3 x^2}-\frac {\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}+\frac {2}{3} a^3 \text {Si}(2 a x)+\frac {16}{3} a^3 \text {Si}(4 a x) \]
a^2/x+cos(a*x)^2/x^3-10*a^2*cos(a*x)^2/x+cos(a*x)^4/a^2/x^5-4/3*cos(a*x)^4 /x^3+32/3*a^2*cos(a*x)^4/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-cos(a*x)^3*sin(a*x)/a/x^4+8/3*a*cos(a*x)^3*sin(a*x)/x^2-cos( a*x)^5/a^2/x^5/(cos(a*x)+a*x*sin(a*x))
Time = 1.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {-10 \cos (a x)+12 a^2 x^2 \cos (a x)-5 \cos (3 a x)+44 a^2 x^2 \cos (3 a x)-\cos (5 a x)+24 a^2 x^2 \cos (5 a x)+8 a x \sin (a x)-8 a^3 x^3 \sin (a x)+12 a x \sin (3 a x)-24 a^3 x^3 \sin (3 a x)+4 a x \sin (5 a x)+32 a^3 x^3 \sin (5 a x)+32 a^3 x^3 (\cos (a x)+a x \sin (a x)) \text {Si}(2 a x)+256 a^3 x^3 (\cos (a x)+a x \sin (a x)) \text {Si}(4 a x)}{48 x^3 (\cos (a x)+a x \sin (a x))} \]
(-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]))
Time = 0.91 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.39, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5110, 3042, 3795, 3042, 3795, 15, 3042, 3794, 27, 2009, 3042, 3780}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cos ^6(a x)}{x^4 (a x \sin (a x)+\cos (a x))^2} \, dx\) |
\(\Big \downarrow \) 5110 |
\(\displaystyle -\frac {5 \int \frac {\cos ^4(a x)}{x^6}dx}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^4}{x^6}dx}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3795 |
\(\displaystyle -\frac {5 \left (-\frac {4}{5} a^2 \int \frac {\cos ^4(a x)}{x^4}dx+\frac {3}{5} a^2 \int \frac {\cos ^2(a x)}{x^4}dx-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^2}{x^4}dx-\frac {4}{5} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^4}{x^4}dx-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3795 |
\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \left (\frac {1}{3} a^2 \int \frac {1}{x^2}dx-\frac {2}{3} a^2 \int \frac {\cos ^2(a x)}{x^2}dx-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (-\frac {8}{3} a^2 \int \frac {\cos ^4(a x)}{x^2}dx+2 a^2 \int \frac {\cos ^2(a x)}{x^2}dx-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \int \frac {\cos ^2(a x)}{x^2}dx-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (-\frac {8}{3} a^2 \int \frac {\cos ^4(a x)}{x^2}dx+2 a^2 \int \frac {\cos ^2(a x)}{x^2}dx-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^2}{x^2}dx-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (2 a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^2}{x^2}dx-\frac {8}{3} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^4}{x^2}dx-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3794 |
\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \left (2 a \int -\frac {\sin (2 a x)}{2 x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (-\frac {8}{3} a^2 \left (4 a \int \left (-\frac {\sin (2 a x)}{4 x}-\frac {\sin (4 a x)}{8 x}\right )dx-\frac {\cos ^4(a x)}{x}\right )+2 a^2 \left (2 a \int -\frac {\sin (2 a x)}{2 x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (-\frac {8}{3} a^2 \left (4 a \int \left (-\frac {\sin (2 a x)}{4 x}-\frac {\sin (4 a x)}{8 x}\right )dx-\frac {\cos ^4(a x)}{x}\right )+2 a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5 \left (-\frac {4}{5} a^2 \left (2 a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {8}{3} a^2 \left (4 a \left (-\frac {1}{4} \text {Si}(2 a x)-\frac {\text {Si}(4 a x)}{8}\right )-\frac {\cos ^4(a x)}{x}\right )-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )+\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5 \left (-\frac {4}{5} a^2 \left (2 a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {8}{3} a^2 \left (4 a \left (-\frac {1}{4} \text {Si}(2 a x)-\frac {\text {Si}(4 a x)}{8}\right )-\frac {\cos ^4(a x)}{x}\right )-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )+\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \left (-a \text {Si}(2 a x)-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (-\frac {8}{3} a^2 \left (4 a \left (-\frac {1}{4} \text {Si}(2 a x)-\frac {\text {Si}(4 a x)}{8}\right )-\frac {\cos ^4(a x)}{x}\right )+2 a^2 \left (-a \text {Si}(2 a x)-\frac {\cos ^2(a x)}{x}\right )-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\) |
-(Cos[a*x]^5/(a^2*x^5*(Cos[a*x] + a*x*Sin[a*x]))) - (5*(-1/5*Cos[a*x]^4/x^ 5 + (a*Cos[a*x]^3*Sin[a*x])/(5*x^4) + (3*a^2*(-1/3*a^2/x - Cos[a*x]^2/(3*x ^3) + (a*Cos[a*x]*Sin[a*x])/(3*x^2) - (2*a^2*(-(Cos[a*x]^2/x) - a*SinInteg ral[2*a*x]))/3))/5 - (4*a^2*(-1/3*Cos[a*x]^4/x^3 + (2*a*Cos[a*x]^3*Sin[a*x ])/(3*x^2) + 2*a^2*(-(Cos[a*x]^2/x) - a*SinIntegral[2*a*x]) - (8*a^2*(-(Co s[a*x]^4/x) + 4*a*(-1/4*SinIntegral[2*a*x] - SinIntegral[4*a*x]/8)))/3))/5 ))/a^2
3.6.94.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), 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] + Simp[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] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
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] - Simp[b^2*((n - 1)/d^2) Int [(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]
Timed out.
\[\int \frac {\cos \left (a x \right )^{6}}{x^{4} \left (\cos \left (a x \right )+a x \sin \left (a x \right )\right )^{2}}d x\]
Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {19 \, a^{2} x^{2} \cos \left (a x\right )^{3} - {\left (24 \, a^{2} x^{2} - 1\right )} \cos \left (a x\right )^{5} - 2 \, {\left (8 \, a^{3} x^{3} \operatorname {Si}\left (4 \, a x\right ) + a^{3} x^{3} \operatorname {Si}\left (2 \, a x\right )\right )} \cos \left (a x\right ) - {\left (16 \, a^{4} x^{4} \operatorname {Si}\left (4 \, a x\right ) + 2 \, a^{4} x^{4} \operatorname {Si}\left (2 \, a x\right ) - 30 \, a^{3} x^{3} \cos \left (a x\right )^{2} + 3 \, a^{3} x^{3} + 4 \, {\left (8 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{4}\right )} \sin \left (a x\right )}{3 \, {\left (a x^{4} \sin \left (a x\right ) + x^{3} \cos \left (a x\right )\right )}} \]
-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*s in_integral(4*a*x) + a^3*x^3*sin_integral(2*a*x))*cos(a*x) - (16*a^4*x^4*s in_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))
\[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {\cos ^{6}{\left (a x \right )}}{x^{4} \left (a x \sin {\left (a x \right )} + \cos {\left (a x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.71 (sec) , antiderivative size = 7279, normalized size of antiderivative = 41.36 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Too large to display} \]
1/12*(64*a^8*x^8*imag_part(cos_integral(4*a*x))*tan(2*a*x)^2*tan(a*x)^2*ta n(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_integr al(-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))*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) - 8*a^8*x^8*imag_part(cos_i ntegral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x) - 64*a^8*x^8*imag_part(cos_inte gral(-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*ta n(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_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)...
Timed out. \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {{\cos \left (a\,x\right )}^6}{x^4\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]