Integrand size = 24, antiderivative size = 80 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {1}{x}+\frac {\cos ^2(a x)}{a^2 x^3}-\frac {2 \cos ^2(a x)}{x}-\frac {\cos (a x) \sin (a x)}{a x^2}-\frac {\cos ^3(a x)}{a^2 x^3 (\cos (a x)+a x \sin (a x))}-2 a \text {Si}(2 a x) \]
1/x+cos(a*x)^2/a^2/x^3-2*cos(a*x)^2/x-2*a*Si(2*a*x)-cos(a*x)*sin(a*x)/a/x^ 2-cos(a*x)^3/a^2/x^3/(cos(a*x)+a*x*sin(a*x))
Time = 1.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {3 \cos (a x)+\cos (3 a x)-2 a x \sin (a x)+2 a x \sin (3 a x)+8 a x (\cos (a x)+a x \sin (a x)) \text {Si}(2 a x)}{4 x (\cos (a x)+a x \sin (a x))} \]
-1/4*(3*Cos[a*x] + Cos[3*a*x] - 2*a*x*Sin[a*x] + 2*a*x*Sin[3*a*x] + 8*a*x* (Cos[a*x] + a*x*Sin[a*x])*SinIntegral[2*a*x])/(x*(Cos[a*x] + a*x*Sin[a*x]) )
Time = 0.50 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5110, 3042, 3795, 15, 3042, 3794, 27, 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.
| \(\displaystyle \int \frac {\cos ^4(a x)}{x^2 (a x \sin (a x)+\cos (a x))^2} \, dx\) |
| \(\Big \downarrow \) 5110 |
| \(\displaystyle -\frac {3 \int \frac {\cos ^2(a x)}{x^4}dx}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^2}{x^4}dx}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {3 \left (\frac {1}{3} a^2 \int \frac {1}{x^2}dx-\frac {2}{3} a^2 \int \frac {\cos ^2(a x)}{x^2}dx-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3 \left (-\frac {2}{3} a^2 \int \frac {\cos ^2(a x)}{x^2}dx-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {2}{3} a^2 \int \frac {\sin \left (a x+\frac {\pi }{2}\right )^2}{x^2}dx-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle -\frac {3 \left (-\frac {2}{3} a^2 \left (2 a \int -\frac {\sin (2 a x)}{2 x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (-\frac {2}{3} a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {2}{3} a^2 \left (-a \int \frac {\sin (2 a x)}{x}dx-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle -\frac {3 \left (-\frac {2}{3} a^2 \left (-a \text {Si}(2 a x)-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\) |
-(Cos[a*x]^3/(a^2*x^3*(Cos[a*x] + a*x*Sin[a*x]))) - (3*(-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*SinIntegral[2*a*x]))/3))/a^2
3.6.96.3.1 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.
\[\int \frac {\cos \left (a x \right )^{4}}{x^{2} \left (\cos \left (a x \right )+a x \sin \left (a x \right )\right )^{2}}d x\]
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 \, a x \cos \left (a x\right ) \operatorname {Si}\left (2 \, a x\right ) + \cos \left (a x\right )^{3} + {\left (2 \, a^{2} x^{2} \operatorname {Si}\left (2 \, a x\right ) + 2 \, a x \cos \left (a x\right )^{2} - a x\right )} \sin \left (a x\right )}{a x^{2} \sin \left (a x\right ) + x \cos \left (a x\right )} \]
-(2*a*x*cos(a*x)*sin_integral(2*a*x) + cos(a*x)^3 + (2*a^2*x^2*sin_integra l(2*a*x) + 2*a*x*cos(a*x)^2 - a*x)*sin(a*x))/(a*x^2*sin(a*x) + x*cos(a*x))
\[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {\cos ^{4}{\left (a x \right )}}{x^{2} \left (a x \sin {\left (a x \right )} + \cos {\left (a x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.40 (sec) , antiderivative size = 997, normalized size of antiderivative = 12.46 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Too large to display} \]
-(2*a^4*x^4*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 2*a^4 *x^4*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x) + 4*a^4*x^4*s in_integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x) - a^3*x^3*imag_part(cos_integra l(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + a^3*x^3*imag_part(cos_integral(-2*a* x))*tan(a*x)^2*tan(1/2*a*x)^2 - 2*a^3*x^3*sin_integral(2*a*x)*tan(a*x)^2*t an(1/2*a*x)^2 + 2*a^4*x^4*imag_part(cos_integral(2*a*x))*tan(1/2*a*x) - 2* a^4*x^4*imag_part(cos_integral(-2*a*x))*tan(1/2*a*x) + 4*a^4*x^4*sin_integ ral(2*a*x)*tan(1/2*a*x) + a^3*x^3*imag_part(cos_integral(2*a*x))*tan(a*x)^ 2 - a^3*x^3*imag_part(cos_integral(-2*a*x))*tan(a*x)^2 + 2*a^3*x^3*sin_int egral(2*a*x)*tan(a*x)^2 - 2*a^3*x^3*tan(a*x)^2*tan(1/2*a*x) - a^3*x^3*imag _part(cos_integral(2*a*x))*tan(1/2*a*x)^2 + a^3*x^3*imag_part(cos_integral (-2*a*x))*tan(1/2*a*x)^2 - 2*a^3*x^3*sin_integral(2*a*x)*tan(1/2*a*x)^2 + 2*a^2*x^2*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x) - 2*a^2*x ^2*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x) + 4*a^2*x^2*sin _integral(2*a*x)*tan(a*x)^2*tan(1/2*a*x) + a^2*x^2*tan(a*x)^2*tan(1/2*a*x) ^2 + a^3*x^3*imag_part(cos_integral(2*a*x)) - a^3*x^3*imag_part(cos_integr al(-2*a*x)) + 2*a^3*x^3*sin_integral(2*a*x) + 2*a^3*x^3*tan(1/2*a*x) - a*x *imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 + a*x*imag_part( cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*x)^2 - 2*a*x*sin_integral(2*a*x )*tan(a*x)^2*tan(1/2*a*x)^2 - a^2*x^2*tan(a*x)^2 + 2*a^2*x^2*imag_part(...
Timed out. \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {{\cos \left (a\,x\right )}^4}{x^2\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]