\(\int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx\) [537]

Optimal result

Integrand size = 24, antiderivative size = 176 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a^2}{x}+\frac {\cos ^2(a x)}{x^3}-\frac {10 a^2 \cos ^2(a x)}{x}+\frac {\cos ^4(a x)}{a^2 x^5}-\frac {4 \cos ^4(a x)}{3 x^3}+\frac {32 a^2 \cos ^4(a x)}{3 x}-\frac {a \cos (a x) \sin (a x)}{x^2}-\frac {\cos ^3(a x) \sin (a x)}{a x^4}+\frac {8 a \cos ^3(a x) \sin (a x)}{3 x^2}-\frac {\cos ^5(a x)}{a^2 x^5 (\cos (a x)+a x \sin (a x))}+\frac {2}{3} a^3 \text {Si}(2 a x)+\frac {16}{3} a^3 \text {Si}(4 a x) \] Output:

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-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) 
)+2/3*a^3*Si(2*a*x)+16/3*a^3*Si(4*a*x)
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {-10 \cos (a x)+12 a^2 x^2 \cos (a x)-5 \cos (3 a x)+44 a^2 x^2 \cos (3 a x)-\cos (5 a x)+24 a^2 x^2 \cos (5 a x)+8 a x \sin (a x)-8 a^3 x^3 \sin (a x)+12 a x \sin (3 a x)-24 a^3 x^3 \sin (3 a x)+4 a x \sin (5 a x)+32 a^3 x^3 \sin (5 a x)+32 a^3 x^3 (\cos (a x)+a x \sin (a x)) \text {Si}(2 a x)+256 a^3 x^3 (\cos (a x)+a x \sin (a x)) \text {Si}(4 a x)}{48 x^3 (\cos (a x)+a x \sin (a x))} \] Input:

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

Output:

(-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]))
 

Rubi [A] (verified)

Time = 0.96 (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

\(\displaystyle -\frac {5 \left (\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \left (-a \text {Si}(2 a x)-\frac {\cos ^2(a x)}{x}\right )-\frac {a^2}{3 x}-\frac {\cos ^2(a x)}{3 x^3}+\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (-\frac {8}{3} a^2 \left (4 a \left (-\frac {1}{4} \text {Si}(2 a x)-\frac {\text {Si}(4 a x)}{8}\right )-\frac {\cos ^4(a x)}{x}\right )+2 a^2 \left (-a \text {Si}(2 a x)-\frac {\cos ^2(a x)}{x}\right )-\frac {\cos ^4(a x)}{3 x^3}+\frac {2 a \sin (a x) \cos ^3(a x)}{3 x^2}\right )-\frac {\cos ^4(a x)}{5 x^5}+\frac {a \sin (a x) \cos ^3(a x)}{5 x^4}\right )}{a^2}-\frac {\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}\)

Input:

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

Output:

-(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
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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] (warning: unable to verify)

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

Time = 2.31 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.89

method	result	size
risch	\(\frac {i a^{3} \operatorname {expIntegral}_{1}\left (-2 i a x \right )}{3}+\frac {5 a^{2} {\mathrm e}^{2 i a x}}{12 x}+\frac {16 a^{3} \operatorname {Si}\left (4 a x \right )}{3}-\frac {3 a^{2}}{8 x}-\frac {1}{8 x^{3}}-\frac {4 x \cos \left (4 a x \right ) a^{4}}{3 \left (-a^{2} x^{2}-1\right )}-\frac {7 \cos \left (4 a x \right ) a^{2}}{6 x \left (-a^{2} x^{2}-1\right )}-\frac {5 a \sin \left (4 a x \right )}{24 x^{2} \left (-a^{2} x^{2}-1\right )}-\frac {{\mathrm e}^{-2 i a x}}{12 x^{3} \left (-a^{2} x^{2}+2 i a x +1\right )}-\frac {i a^{3} \operatorname {expIntegral}_{1}\left (2 i a x \right )}{3}+\frac {3 i a^{3}}{8 \left (a x +i\right )^{2}}+\frac {\cos \left (4 a x \right )}{24 x^{3} \left (-a^{2} x^{2}-1\right )}-\frac {a^{3} \sin \left (4 a x \right )}{3 \left (-a^{2} x^{2}-1\right )}-\frac {8 \pi \,\operatorname {csgn}\left (a x \right ) a^{3}}{3}-\frac {a^{3}}{4 \left (a x +i\right )^{3}}-\frac {{\mathrm e}^{2 i a x}}{12 x^{3}}+\frac {a^{2} {\mathrm e}^{-2 i a x}}{12 x \left (-a^{2} x^{2}+2 i a x +1\right )}-\frac {5 i a \,{\mathrm e}^{2 i a x}}{24 x^{2}}-\frac {i a^{3} {\mathrm e}^{2 i a x}}{4 \left (i a x -1\right )}+\frac {3 a^{3}}{8 \left (a x +i\right )}+\frac {i a^{3} {\mathrm e}^{-2 i a x}}{-4 a^{2} x^{2}+8 i a x +4}-\frac {2 i a^{3}}{\left (a^{4} x^{4}+2 i a^{3} x^{3}+2 i a x -1\right ) \left (a x -i\right ) \left ({\mathrm e}^{2 i a x} a x -a x +i {\mathrm e}^{2 i a x}+i\right )}+\frac {i a \,{\mathrm e}^{-2 i a x}}{24 x^{2} \left (-a^{2} x^{2}+2 i a x +1\right )}-\frac {a^{4} {\mathrm e}^{-2 i a x} x}{6 \left (-a^{2} x^{2}+2 i a x +1\right )}+\frac {i a^{3} {\mathrm e}^{2 i a x}}{8 \left (i a x -1\right )^{2}}\)	\(509\)

Input:

int(cos(a*x)^6/x^4/(cos(a*x)+a*x*sin(a*x))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/12*exp(-2*I*a*x)/x^3/(-a^2*x^2+2*I*a*x+1)+3/8*a^3/(a*x+I)-1/4*a^3/(a*x+ 
I)^3+16/3*a^3*Si(4*a*x)-3/8*a^2/x-1/8/x^3+5/12*a^2/x*exp(2*I*a*x)-1/6*a^4* 
exp(-2*I*a*x)*x/(-a^2*x^2+2*I*a*x+1)+1/12*a^2*exp(-2*I*a*x)/x/(-a^2*x^2+2* 
I*a*x+1)-4/3*x/(-a^2*x^2-1)*cos(4*a*x)*a^4-7/6/x/(-a^2*x^2-1)*cos(4*a*x)*a 
^2-5/24*a/x^2/(-a^2*x^2-1)*sin(4*a*x)-2*I*a^3/(a^4*x^4-1+2*I*a*x+2*I*a^3*x 
^3)/(a*x-I)/(exp(2*I*a*x)*a*x-a*x+I*exp(2*I*a*x)+I)-1/12/x^3*exp(2*I*a*x)+ 
1/24/x^3/(-a^2*x^2-1)*cos(4*a*x)-1/3*a^3/(-a^2*x^2-1)*sin(4*a*x)-8/3*Pi*cs 
gn(a*x)*a^3+1/3*I*a^3*Ei(1,-2*I*a*x)-1/3*I*a^3*Ei(1,2*I*a*x)+3/8*I*a^3/(a* 
x+I)^2+1/24*I*a*exp(-2*I*a*x)/x^2/(-a^2*x^2+2*I*a*x+1)-5/24*I*a/x^2*exp(2* 
I*a*x)-1/4*I*a^3*exp(2*I*a*x)/(I*a*x-1)+1/8*I*a^3*exp(2*I*a*x)/(I*a*x-1)^2 
+1/4*I*a^3*exp(-2*I*a*x)/(-a^2*x^2+2*I*a*x+1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {19 \, a^{2} x^{2} \cos \left (a x\right )^{3} - {\left (24 \, a^{2} x^{2} - 1\right )} \cos \left (a x\right )^{5} - 2 \, {\left (8 \, a^{3} x^{3} \operatorname {Si}\left (4 \, a x\right ) + a^{3} x^{3} \operatorname {Si}\left (2 \, a x\right )\right )} \cos \left (a x\right ) - {\left (16 \, a^{4} x^{4} \operatorname {Si}\left (4 \, a x\right ) + 2 \, a^{4} x^{4} \operatorname {Si}\left (2 \, a x\right ) - 30 \, a^{3} x^{3} \cos \left (a x\right )^{2} + 3 \, a^{3} x^{3} + 4 \, {\left (8 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{4}\right )} \sin \left (a x\right )}{3 \, {\left (a x^{4} \sin \left (a x\right ) + x^{3} \cos \left (a x\right )\right )}} \] Input:

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

Output:

-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))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(cos(a*x)^6/x^4/(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 = 0.71 (sec) , antiderivative size = 7279, normalized size of antiderivative = 41.36 \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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)...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {{\cos \left (a\,x\right )}^6}{x^4\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {\cos \left (a x \right )^{6}}{\cos \left (a x \right )^{2} x^{4}+2 \cos \left (a x \right ) \sin \left (a x \right ) a \,x^{5}+\sin \left (a x \right )^{2} a^{2} x^{6}}d x \] Input:

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

Output:

int(cos(a*x)**6/(cos(a*x)**2*x**4 + 2*cos(a*x)*sin(a*x)*a*x**5 + sin(a*x)* 
*2*a**2*x**6),x)