\(\int \frac {\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx\) [528]

Optimal result

Integrand size = 26, antiderivative size = 175 \[ \int \frac {\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {a^2}{x}+\frac {a \cos (a x) \sin (a x)}{x^2}+\frac {\sin ^2(a x)}{x^3}-\frac {10 a^2 \sin ^2(a x)}{x}+\frac {\cos (a x) \sin ^3(a x)}{a x^4}-\frac {8 a \cos (a x) \sin ^3(a x)}{3 x^2}+\frac {\sin ^4(a x)}{a^2 x^5}-\frac {4 \sin ^4(a x)}{3 x^3}+\frac {32 a^2 \sin ^4(a x)}{3 x}+\frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}-\frac {2}{3} a^3 \text {Si}(2 a x)+\frac {16}{3} a^3 \text {Si}(4 a x) \] Output:

a^2/x+a*cos(a*x)*sin(a*x)/x^2+sin(a*x)^2/x^3-10*a^2*sin(a*x)^2/x+cos(a*x)* 
sin(a*x)^3/a/x^4-8/3*a*cos(a*x)*sin(a*x)^3/x^2+sin(a*x)^4/a^2/x^5-4/3*sin( 
a*x)^4/x^3+32/3*a^2*sin(a*x)^4/x+sin(a*x)^5/a^2/x^5/(a*x*cos(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 = 1.46 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {8 a x \cos (a x)-8 a^3 x^3 \cos (a x)-12 a x \cos (3 a x)+24 a^3 x^3 \cos (3 a x)+4 a x \cos (5 a x)+32 a^3 x^3 \cos (5 a x)+10 \sin (a x)-12 a^2 x^2 \sin (a x)-5 \sin (3 a x)+44 a^2 x^2 \sin (3 a x)+\sin (5 a x)-24 a^2 x^2 \sin (5 a x)-32 a^3 x^3 (a x \cos (a x)-\sin (a x)) \text {Si}(2 a x)+256 a^3 x^3 (a x \cos (a x)-\sin (a x)) \text {Si}(4 a x)}{48 x^3 (a x \cos (a x)-\sin (a x))} \] Input:

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

Output:

(8*a*x*Cos[a*x] - 8*a^3*x^3*Cos[a*x] - 12*a*x*Cos[3*a*x] + 24*a^3*x^3*Cos[ 
3*a*x] + 4*a*x*Cos[5*a*x] + 32*a^3*x^3*Cos[5*a*x] + 10*Sin[a*x] - 12*a^2*x 
^2*Sin[a*x] - 5*Sin[3*a*x] + 44*a^2*x^2*Sin[3*a*x] + Sin[5*a*x] - 24*a^2*x 
^2*Sin[5*a*x] - 32*a^3*x^3*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[2*a*x] + 
256*a^3*x^3*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[4*a*x])/(48*x^3*(a*x*Cos 
[a*x] - Sin[a*x]))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.39, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5109, 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.

\(\displaystyle \int \frac {\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx\)

\(\Big \downarrow \) 5109

\(\displaystyle \frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}-\frac {5 \int \frac {\sin ^4(a x)}{x^6}dx}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}-\frac {5 \int \frac {\sin (a x)^4}{x^6}dx}{a^2}\)

\(\Big \downarrow \) 3795

\(\displaystyle \frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}-\frac {5 \left (-\frac {4}{5} a^2 \int \frac {\sin ^4(a x)}{x^4}dx+\frac {3}{5} a^2 \int \frac {\sin ^2(a x)}{x^4}dx-\frac {\sin ^4(a x)}{5 x^5}-\frac {a \sin ^3(a x) \cos (a x)}{5 x^4}\right )}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}-\frac {5 \left (\frac {3}{5} a^2 \int \frac {\sin (a x)^2}{x^4}dx-\frac {4}{5} a^2 \int \frac {\sin (a x)^4}{x^4}dx-\frac {\sin ^4(a x)}{5 x^5}-\frac {a \sin ^3(a x) \cos (a x)}{5 x^4}\right )}{a^2}\)

\(\Big \downarrow \) 3795

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}-\frac {5 \left (\frac {3}{5} a^2 \left (-\frac {2}{3} a^2 \int \frac {\sin (a x)^2}{x^2}dx-\frac {a^2}{3 x}-\frac {\sin ^2(a x)}{3 x^3}-\frac {a \sin (a x) \cos (a x)}{3 x^2}\right )-\frac {4}{5} a^2 \left (2 a^2 \int \frac {\sin (a x)^2}{x^2}dx-\frac {8}{3} a^2 \int \frac {\sin (a x)^4}{x^2}dx-\frac {\sin ^4(a x)}{3 x^3}-\frac {2 a \sin ^3(a x) \cos (a x)}{3 x^2}\right )-\frac {\sin ^4(a x)}{5 x^5}-\frac {a \sin ^3(a x) \cos (a x)}{5 x^4}\right )}{a^2}\)

\(\Big \downarrow \) 3794

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

Input:

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

Output:

Sin[a*x]^5/(a^2*x^5*(a*x*Cos[a*x] - Sin[a*x])) - (5*(-1/5*(a*Cos[a*x]*Sin[ 
a*x]^3)/x^4 - Sin[a*x]^4/(5*x^5) + (3*a^2*(-1/3*a^2/x - (a*Cos[a*x]*Sin[a* 
x])/(3*x^2) - Sin[a*x]^2/(3*x^3) - (2*a^2*(-(Sin[a*x]^2/x) + a*SinIntegral 
[2*a*x]))/3))/5 - (4*a^2*((-2*a*Cos[a*x]*Sin[a*x]^3)/(3*x^2) - Sin[a*x]^4/ 
(3*x^3) + 2*a^2*(-(Sin[a*x]^2/x) + a*SinIntegral[2*a*x]) - (8*a^2*(-(Sin[a 
*x]^4/x) + 4*a*(SinIntegral[2*a*x]/4 - 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 5109

Int[(((b_.)*(x_))^(m_)*Sin[(a_.)*(x_)]^(n_))/(Cos[(a_.)*(x_)]*(d_.)*(x_) + 
(c_.)*Sin[(a_.)*(x_)])^2, x_Symbol] :> Simp[b*(b*x)^(m - 1)*(Sin[a*x]^(n - 
1)/(a*d*(c*Sin[a*x] + d*x*Cos[a*x]))), x] - Simp[b^2*((n - 1)/d^2)   Int[(b 
*x)^(m - 2)*Sin[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 = 2.81 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.90

method	result	size
risch	\(\frac {{\mathrm e}^{-2 i a x}}{12 x^{3} \left (-a^{2} x^{2}+2 i a x +1\right )}-\frac {i a^{3} \operatorname {expIntegral}_{1}\left (-2 i a x \right )}{3}+\frac {i a^{3} \operatorname {expIntegral}_{1}\left (2 i a x \right )}{3}-\frac {5 a^{2} {\mathrm e}^{2 i a x}}{12 x}+\frac {3 i a^{3}}{8 \left (a x +i\right )^{2}}+\frac {16 a^{3} \operatorname {Si}\left (4 a x \right )}{3}-\frac {3 a^{2}}{8 x}-\frac {1}{8 x^{3}}-\frac {4 x \cos \left (4 a x \right ) a^{4}}{3 \left (-a^{2} x^{2}-1\right )}-\frac {7 \cos \left (4 a x \right ) a^{2}}{6 x \left (-a^{2} x^{2}-1\right )}-\frac {5 a \sin \left (4 a x \right )}{24 x^{2} \left (-a^{2} x^{2}-1\right )}+\frac {\cos \left (4 a x \right )}{24 x^{3} \left (-a^{2} x^{2}-1\right )}-\frac {a^{3} \sin \left (4 a x \right )}{3 \left (-a^{2} x^{2}-1\right )}-\frac {8 \pi \,\operatorname {csgn}\left (a x \right ) a^{3}}{3}+\frac {3 a^{3}}{8 \left (a x +i\right )}-\frac {a^{3}}{4 \left (a x +i\right )^{3}}+\frac {{\mathrm e}^{2 i a x}}{12 x^{3}}+\frac {a^{4} {\mathrm e}^{-2 i a x} x}{-6 a^{2} x^{2}+12 i a x +6}-\frac {a^{2} {\mathrm e}^{-2 i a x}}{12 x \left (-a^{2} x^{2}+2 i a x +1\right )}+\frac {5 i a \,{\mathrm e}^{2 i a x}}{24 x^{2}}-\frac {i a^{3} {\mathrm e}^{2 i a x}}{8 \left (i a x -1\right )^{2}}-\frac {i a^{3} {\mathrm e}^{-2 i a x}}{4 \left (-a^{2} x^{2}+2 i a x +1\right )}+\frac {i a^{3} {\mathrm e}^{2 i a x}}{4 i a x -4}+\frac {2 i a^{3}}{\left (a^{4} x^{4}+2 i a^{3} x^{3}+2 i a x -1\right ) \left (a x -i\right ) \left ({\mathrm e}^{2 i a x} a x +i {\mathrm e}^{2 i a x}+a x -i\right )}-\frac {i a \,{\mathrm e}^{-2 i a x}}{24 x^{2} \left (-a^{2} x^{2}+2 i a x +1\right )}\)	\(508\)

Input:

int(sin(a*x)^6/x^4/(a*x*cos(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)-1/3*I*a^3*Ei(1,-2*I*a*x)+1/3*I 
*a^3*Ei(1,2*I*a*x)+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)+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)+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*csgn(a*x)*a^3+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+I*exp(2*I*a*x)+a*x-I)+3/8*I*a^3/(a*x+I)^2-1/2 
4*I*a*exp(-2*I*a*x)/x^2/(-a^2*x^2+2*I*a*x+1)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {4 \, {\left (8 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{5} - 2 \, {\left (17 \, a^{3} x^{3} + 4 \, a x\right )} \cos \left (a x\right )^{3} + {\left (16 \, a^{4} x^{4} \operatorname {Si}\left (4 \, a x\right ) - 2 \, a^{4} x^{4} \operatorname {Si}\left (2 \, a x\right ) + 5 \, a^{3} x^{3} + 4 \, a x\right )} \cos \left (a x\right ) - {\left (16 \, a^{3} x^{3} \operatorname {Si}\left (4 \, a x\right ) - 2 \, a^{3} x^{3} \operatorname {Si}\left (2 \, a x\right ) + {\left (24 \, a^{2} x^{2} - 1\right )} \cos \left (a x\right )^{4} + 5 \, a^{2} x^{2} - {\left (29 \, a^{2} x^{2} - 2\right )} \cos \left (a x\right )^{2} - 1\right )} \sin \left (a x\right )}{3 \, {\left (a x^{4} \cos \left (a x\right ) - x^{3} \sin \left (a x\right )\right )}} \] Input:

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

Output:

1/3*(4*(8*a^3*x^3 + a*x)*cos(a*x)^5 - 2*(17*a^3*x^3 + 4*a*x)*cos(a*x)^3 + 
(16*a^4*x^4*sin_integral(4*a*x) - 2*a^4*x^4*sin_integral(2*a*x) + 5*a^3*x^ 
3 + 4*a*x)*cos(a*x) - (16*a^3*x^3*sin_integral(4*a*x) - 2*a^3*x^3*sin_inte 
gral(2*a*x) + (24*a^2*x^2 - 1)*cos(a*x)^4 + 5*a^2*x^2 - (29*a^2*x^2 - 2)*c 
os(a*x)^2 - 1)*sin(a*x))/(a*x^4*cos(a*x) - x^3*sin(a*x))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(sin(a*x)^6/x^4/(a*x*cos(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 = 0.78 (sec) , antiderivative size = 7347, normalized size of antiderivative = 41.98 \[ \int \frac {\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(32*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)^2 - 4*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)^2 + 4*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)^2 - 32*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)^2 + 64*a^8*x^8*sin_integral(4*a*x)*t 
an(2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 8*a^8*x^8*sin_integral(2*a*x)*tan( 
2*a*x)^2*tan(a*x)^2*tan(1/2*a*x)^2 - 32*a^8*x^8*imag_part(cos_integral(4*a 
*x))*tan(2*a*x)^2*tan(a*x)^2 + 4*a^8*x^8*imag_part(cos_integral(2*a*x))*ta 
n(2*a*x)^2*tan(a*x)^2 - 4*a^8*x^8*imag_part(cos_integral(-2*a*x))*tan(2*a* 
x)^2*tan(a*x)^2 + 32*a^8*x^8*imag_part(cos_integral(-4*a*x))*tan(2*a*x)^2* 
tan(a*x)^2 - 64*a^8*x^8*sin_integral(4*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 8*a^ 
8*x^8*sin_integral(2*a*x)*tan(2*a*x)^2*tan(a*x)^2 + 32*a^8*x^8*imag_part(c 
os_integral(4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 4*a^8*x^8*imag_part(cos_ 
integral(2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 4*a^8*x^8*imag_part(cos_int 
egral(-2*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 - 32*a^8*x^8*imag_part(cos_inte 
gral(-4*a*x))*tan(2*a*x)^2*tan(1/2*a*x)^2 + 64*a^8*x^8*sin_integral(4*a*x) 
*tan(2*a*x)^2*tan(1/2*a*x)^2 - 8*a^8*x^8*sin_integral(2*a*x)*tan(2*a*x)^2* 
tan(1/2*a*x)^2 + 32*a^8*x^8*imag_part(cos_integral(4*a*x))*tan(a*x)^2*tan( 
1/2*a*x)^2 - 4*a^8*x^8*imag_part(cos_integral(2*a*x))*tan(a*x)^2*tan(1/2*a 
*x)^2 + 4*a^8*x^8*imag_part(cos_integral(-2*a*x))*tan(a*x)^2*tan(1/2*a*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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