\(\int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx\) [539]

Optimal result

Integrand size = 24, antiderivative size = 80 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {1}{x}+\frac {\cos ^2(a x)}{a^2 x^3}-\frac {2 \cos ^2(a x)}{x}-\frac {\cos (a x) \sin (a x)}{a x^2}-\frac {\cos ^3(a x)}{a^2 x^3 (\cos (a x)+a x \sin (a x))}-2 a \text {Si}(2 a x) \] Output:

1/x+cos(a*x)^2/a^2/x^3-2*cos(a*x)^2/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))-2*a*Si(2*a*x)
 

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {3 \cos (a x)+\cos (3 a x)-2 a x \sin (a x)+2 a x \sin (3 a x)+8 a x (\cos (a x)+a x \sin (a x)) \text {Si}(2 a x)}{4 x (\cos (a x)+a x \sin (a x))} \] Input:

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

Output:

-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]) 
)
 

Rubi [A] (verified)

Time = 0.52 (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

\(\displaystyle -\frac {3 \left (-\frac {2}{3} a^2 \left (-a \text {Si}(2 a x)-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )}{a^2}-\frac {\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}\)

Input:

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

Output:

-(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
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

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]
 

rule 3794

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]
 

rule 3795

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]
 

rule 5110

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.46

method	result	size
risch	\(\frac {a}{2 a x +2 i}+\frac {i a}{2 \left (a x +i\right )^{2}}-\frac {1}{2 x}+\frac {i \left (-4 i \operatorname {expIntegral}_{1}\left (-2 i a x \right ) a^{2} x^{2}-2 \,{\mathrm e}^{2 i a x} a x +4 \,\operatorname {expIntegral}_{1}\left (-2 i a x \right ) a x -i {\mathrm e}^{2 i a x}\right )}{4 \left (i a x -1\right ) x}-\frac {i a \,{\mathrm e}^{-2 i a x}}{2 \left (i a x +1\right )}-\frac {{\mathrm e}^{-2 i a x}}{4 \left (i a x +1\right ) x}+i a \,\operatorname {expIntegral}_{1}\left (2 i a x \right )-\frac {2 i a}{\left (a^{2} x^{2}+2 i a x -1\right ) \left (a x -i\right ) \left ({\mathrm e}^{2 i a x} a x -a x +i {\mathrm e}^{2 i a x}+i\right )}\)	\(197\)

Input:

int(cos(a*x)^4/x^2/(cos(a*x)+a*x*sin(a*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*a/(a*x+I)+1/2*I*a/(a*x+I)^2-1/2/x+1/4*I*(-4*I*Ei(1,-2*I*a*x)*a^2*x^2-2 
*exp(2*I*a*x)*a*x+4*Ei(1,-2*I*a*x)*a*x-I*exp(2*I*a*x))/(I*a*x-1)/x-1/2*I*a 
*exp(-2*I*a*x)/(I*a*x+1)-1/4*exp(-2*I*a*x)/(I*a*x+1)/x+I*a*Ei(1,2*I*a*x)-2 
*I*a/(a^2*x^2+2*I*a*x-1)/(a*x-I)/(exp(2*I*a*x)*a*x-a*x+I*exp(2*I*a*x)+I)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 \, a x \cos \left (a x\right ) \operatorname {Si}\left (2 \, a x\right ) + \cos \left (a x\right )^{3} + {\left (2 \, a^{2} x^{2} \operatorname {Si}\left (2 \, a x\right ) + 2 \, a x \cos \left (a x\right )^{2} - a x\right )} \sin \left (a x\right )}{a x^{2} \sin \left (a x\right ) + x \cos \left (a x\right )} \] Input:

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

Output:

-(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))
 

Sympy [F]

\[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {\cos ^{4}{\left (a x \right )}}{x^{2} \left (a x \sin {\left (a x \right )} + \cos {\left (a x \right )}\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.37 (sec) , antiderivative size = 997, normalized size of antiderivative = 12.46 \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-(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(...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {{\cos \left (a\,x\right )}^4}{x^2\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {\cos \left (a x \right )^{4}}{\cos \left (a x \right )^{2} x^{2}+2 \cos \left (a x \right ) \sin \left (a x \right ) a \,x^{3}+\sin \left (a x \right )^{2} a^{2} x^{4}}d x \] Input:

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

Output:

int(cos(a*x)**4/(cos(a*x)**2*x**2 + 2*cos(a*x)*sin(a*x)*a*x**3 + sin(a*x)* 
*2*a**2*x**4),x)