Integrand size = 24, antiderivative size = 124 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 i x^2}{a^3}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}-\frac {2 i \operatorname {PolyLog}\left (2,-e^{2 i a x}\right )}{a^5}-\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^5}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3} \]
-2*I*x^2/a^3+4*x*ln(1+exp(2*I*a*x))/a^4-2*I*polylog(2,-exp(2*I*a*x))/a^5-x *sec(a*x)^2/a^4-x^3*sec(a*x)^3/a^2/(cos(a*x)+a*x*sin(a*x))+tan(a*x)/a^5+2* x^2*tan(a*x)/a^3+x^2*sec(a*x)^2*tan(a*x)/a^3
Time = 1.83 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {-a x \left (1+2 i a x+a^2 x^2-4 \log \left (1+e^{2 i a x}\right )\right )+\left (1+2 a^2 x^2-2 i a^3 x^3+4 a^2 x^2 \log \left (1+e^{2 i a x}\right )\right ) \tan (a x)+a^3 x^3 \tan ^2(a x)-2 i \operatorname {PolyLog}\left (2,-e^{2 i a x}\right ) (1+a x \tan (a x))}{a^5 (1+a x \tan (a x))} \]
(-(a*x*(1 + (2*I)*a*x + a^2*x^2 - 4*Log[1 + E^((2*I)*a*x)])) + (1 + 2*a^2* x^2 - (2*I)*a^3*x^3 + 4*a^2*x^2*Log[1 + E^((2*I)*a*x)])*Tan[a*x] + a^3*x^3 *Tan[a*x]^2 - (2*I)*PolyLog[2, -E^((2*I)*a*x)]*(1 + a*x*Tan[a*x]))/(a^5*(1 + a*x*Tan[a*x]))
Time = 0.78 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.26, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5112, 3042, 4674, 3042, 4254, 24, 4672, 25, 3042, 4202, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {x^4 \sec ^2(a x)}{(a x \sin (a x)+\cos (a x))^2} \, dx\) |
| \(\Big \downarrow \) 5112 |
| \(\displaystyle \frac {3 \int x^2 \sec ^4(a x)dx}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int x^2 \csc \left (a x+\frac {\pi }{2}\right )^4dx}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {3 \left (\frac {\int \sec ^2(a x)dx}{3 a^2}+\frac {2}{3} \int x^2 \sec ^2(a x)dx-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \csc \left (a x+\frac {\pi }{2}\right )^2dx}{3 a^2}+\frac {2}{3} \int x^2 \csc \left (a x+\frac {\pi }{2}\right )^2dx-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {3 \left (-\frac {\int 1d(-\tan (a x))}{3 a^3}+\frac {2}{3} \int x^2 \csc \left (a x+\frac {\pi }{2}\right )^2dx-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {3 \left (\frac {2}{3} \int x^2 \csc \left (a x+\frac {\pi }{2}\right )^2dx+\frac {\tan (a x)}{3 a^3}-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {3 \left (\frac {2}{3} \left (\frac {2 \int -x \tan (a x)dx}{a}+\frac {x^2 \tan (a x)}{a}\right )+\frac {\tan (a x)}{3 a^3}-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {2}{3} \left (\frac {x^2 \tan (a x)}{a}-\frac {2 \int x \tan (a x)dx}{a}\right )+\frac {\tan (a x)}{3 a^3}-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2}{3} \left (\frac {x^2 \tan (a x)}{a}-\frac {2 \int x \tan (a x)dx}{a}\right )+\frac {\tan (a x)}{3 a^3}-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {3 \left (\frac {2}{3} \left (\frac {x^2 \tan (a x)}{a}-\frac {2 \left (\frac {i x^2}{2}-2 i \int \frac {e^{2 i a x} x}{1+e^{2 i a x}}dx\right )}{a}\right )+\frac {\tan (a x)}{3 a^3}-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {3 \left (\frac {2}{3} \left (\frac {x^2 \tan (a x)}{a}-\frac {2 \left (\frac {i x^2}{2}-2 i \left (\frac {i \int \log \left (1+e^{2 i a x}\right )dx}{2 a}-\frac {i x \log \left (1+e^{2 i a x}\right )}{2 a}\right )\right )}{a}\right )+\frac {\tan (a x)}{3 a^3}-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {3 \left (\frac {2}{3} \left (\frac {x^2 \tan (a x)}{a}-\frac {2 \left (\frac {i x^2}{2}-2 i \left (\frac {\int e^{-2 i a x} \log \left (1+e^{2 i a x}\right )de^{2 i a x}}{4 a^2}-\frac {i x \log \left (1+e^{2 i a x}\right )}{2 a}\right )\right )}{a}\right )+\frac {\tan (a x)}{3 a^3}-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {3 \left (\frac {\tan (a x)}{3 a^3}+\frac {2}{3} \left (\frac {x^2 \tan (a x)}{a}-\frac {2 \left (\frac {i x^2}{2}-2 i \left (-\frac {\operatorname {PolyLog}\left (2,-e^{2 i a x}\right )}{4 a^2}-\frac {i x \log \left (1+e^{2 i a x}\right )}{2 a}\right )\right )}{a}\right )-\frac {x \sec ^2(a x)}{3 a^2}+\frac {x^2 \tan (a x) \sec ^2(a x)}{3 a}\right )}{a^2}\) |
-((x^3*Sec[a*x]^3)/(a^2*(Cos[a*x] + a*x*Sin[a*x]))) + (3*(-1/3*(x*Sec[a*x] ^2)/a^2 + Tan[a*x]/(3*a^3) + (x^2*Sec[a*x]^2*Tan[a*x])/(3*a) + (2*((-2*((I /2)*x^2 - (2*I)*(((-1/2*I)*x*Log[1 + E^((2*I)*a*x)])/a - PolyLog[2, -E^((2 *I)*a*x)]/(4*a^2))))/a + (x^2*Tan[a*x])/a))/3))/a^2
3.7.2.3.1 Defintions of rubi rules used
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Int[(((b_.)*(x_))^(m_.)*Sec[(a_.)*(x_)]^(n_.))/(Cos[(a_.)*(x_)]*(c_.) + (d_ .)*(x_)*Sin[(a_.)*(x_)])^2, x_Symbol] :> Simp[(-b)*(b*x)^(m - 1)*(Sec[a*x]^ (n + 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x]))), x] + Simp[b^2*((n + 1)/d^2) I nt[(b*x)^(m - 2)*Sec[a*x]^(n + 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] & & EqQ[a*c - d, 0] && EqQ[m, n + 2]
Time = 3.54 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(-\frac {2 i \left (-2 i a^{2} x^{2} {\mathrm e}^{2 i a x}+2 a^{3} x^{3}-2 i a^{2} x^{2}+a x \,{\mathrm e}^{2 i a x}-i {\mathrm e}^{2 i a x}+a x -i\right )}{\left (1+{\mathrm e}^{2 i a x}\right ) \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right ) a^{5}}-\frac {4 i x^{2}}{a^{3}}+\frac {4 x \ln \left (1+{\mathrm e}^{2 i a x}\right )}{a^{4}}-\frac {2 i \operatorname {polylog}\left (2, -{\mathrm e}^{2 i a x}\right )}{a^{5}}\) | \(141\) |
-2*I*(-2*I*a^2*x^2*exp(2*I*a*x)+2*a^3*x^3-2*I*a^2*x^2+a*x*exp(2*I*a*x)-I*e xp(2*I*a*x)+a*x-I)/(1+exp(2*I*a*x))/(a*x*exp(2*I*a*x)-a*x+I*exp(2*I*a*x)+I )/a^5-4*I/a^3*x^2+4*x*ln(1+exp(2*I*a*x))/a^4-2*I*polylog(2,-exp(2*I*a*x))/ a^5
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (113) = 226\).
Time = 0.28 (sec) , antiderivative size = 382, normalized size of antiderivative = 3.08 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a^{3} x^{3} - {\left (2 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{2} + {\left (2 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right ) \sin \left (a x\right ) - 2 \, {\left (-i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) - 2 \, {\left (i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) - 2 \, {\left (i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) - 2 \, {\left (-i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (-i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (-i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right )}{a^{6} x \cos \left (a x\right ) \sin \left (a x\right ) + a^{5} \cos \left (a x\right )^{2}} \]
(a^3*x^3 - (2*a^3*x^3 + a*x)*cos(a*x)^2 + (2*a^2*x^2 + 1)*cos(a*x)*sin(a*x ) - 2*(-I*a*x*cos(a*x)*sin(a*x) - I*cos(a*x)^2)*dilog(I*cos(a*x) + sin(a*x )) - 2*(I*a*x*cos(a*x)*sin(a*x) + I*cos(a*x)^2)*dilog(I*cos(a*x) - sin(a*x )) - 2*(I*a*x*cos(a*x)*sin(a*x) + I*cos(a*x)^2)*dilog(-I*cos(a*x) + sin(a* x)) - 2*(-I*a*x*cos(a*x)*sin(a*x) - I*cos(a*x)^2)*dilog(-I*cos(a*x) - sin( a*x)) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2)*log(I*cos(a*x) + si n(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2)*log(I*cos(a*x ) - sin(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2)*log(-I* cos(a*x) + sin(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2)* log(-I*cos(a*x) - sin(a*x) + 1))/(a^6*x*cos(a*x)*sin(a*x) + a^5*cos(a*x)^2 )
\[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {x^{4} \sec ^{2}{\left (a x \right )}}{\left (a x \sin {\left (a x \right )} + \cos {\left (a x \right )}\right )^{2}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (113) = 226\).
Time = 0.32 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.00 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 \, {\left (a x + 2 \, {\left (a^{2} x^{2} - 2 i \, a x \cos \left (2 \, a x\right ) + 2 \, a x \sin \left (2 \, a x\right ) - i \, a x - {\left (a^{2} x^{2} + i \, a x\right )} \cos \left (4 \, a x\right ) + {\left (-i \, a^{2} x^{2} + a x\right )} \sin \left (4 \, a x\right )\right )} \arctan \left (\sin \left (2 \, a x\right ), \cos \left (2 \, a x\right ) + 1\right ) + 2 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2}\right )} \cos \left (4 \, a x\right ) + {\left (2 i \, a^{2} x^{2} + a x - i\right )} \cos \left (2 \, a x\right ) - {\left (a x - {\left (a x + i\right )} \cos \left (4 \, a x\right ) - {\left (i \, a x - 1\right )} \sin \left (4 \, a x\right ) - 2 i \, \cos \left (2 \, a x\right ) + 2 \, \sin \left (2 \, a x\right ) - i\right )} {\rm Li}_2\left (-e^{\left (2 i \, a x\right )}\right ) + {\left (-i \, a^{2} x^{2} - 2 \, a x \cos \left (2 \, a x\right ) - 2 i \, a x \sin \left (2 \, a x\right ) - a x + {\left (i \, a^{2} x^{2} - a x\right )} \cos \left (4 \, a x\right ) - {\left (a^{2} x^{2} + i \, a x\right )} \sin \left (4 \, a x\right )\right )} \log \left (\cos \left (2 \, a x\right )^{2} + \sin \left (2 \, a x\right )^{2} + 2 \, \cos \left (2 \, a x\right ) + 1\right ) + 2 \, {\left (i \, a^{3} x^{3} - a^{2} x^{2}\right )} \sin \left (4 \, a x\right ) - {\left (2 \, a^{2} x^{2} - i \, a x - 1\right )} \sin \left (2 \, a x\right ) - i\right )}}{{\left (i \, a x + {\left (-i \, a x + 1\right )} \cos \left (4 \, a x\right ) + {\left (a x + i\right )} \sin \left (4 \, a x\right ) + 2 \, \cos \left (2 \, a x\right ) + 2 i \, \sin \left (2 \, a x\right ) + 1\right )} a^{5}} \]
-2*(a*x + 2*(a^2*x^2 - 2*I*a*x*cos(2*a*x) + 2*a*x*sin(2*a*x) - I*a*x - (a^ 2*x^2 + I*a*x)*cos(4*a*x) + (-I*a^2*x^2 + a*x)*sin(4*a*x))*arctan2(sin(2*a *x), cos(2*a*x) + 1) + 2*(a^3*x^3 + I*a^2*x^2)*cos(4*a*x) + (2*I*a^2*x^2 + a*x - I)*cos(2*a*x) - (a*x - (a*x + I)*cos(4*a*x) - (I*a*x - 1)*sin(4*a*x ) - 2*I*cos(2*a*x) + 2*sin(2*a*x) - I)*dilog(-e^(2*I*a*x)) + (-I*a^2*x^2 - 2*a*x*cos(2*a*x) - 2*I*a*x*sin(2*a*x) - a*x + (I*a^2*x^2 - a*x)*cos(4*a*x ) - (a^2*x^2 + I*a*x)*sin(4*a*x))*log(cos(2*a*x)^2 + sin(2*a*x)^2 + 2*cos( 2*a*x) + 1) + 2*(I*a^3*x^3 - a^2*x^2)*sin(4*a*x) - (2*a^2*x^2 - I*a*x - 1) *sin(2*a*x) - I)/((I*a*x + (-I*a*x + 1)*cos(4*a*x) + (a*x + I)*sin(4*a*x) + 2*cos(2*a*x) + 2*I*sin(2*a*x) + 1)*a^5)
\[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int { \frac {x^{4} \sec \left (a x\right )^{2}}{{\left (a x \sin \left (a x\right ) + \cos \left (a x\right )\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {x^4}{{\cos \left (a\,x\right )}^2\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]