SIMPLE;

WRITE it as

integr. (e^logx^9 -e^logx^5)/(e^logx^2 - e^ logx^4) dx      [using property alogx =logx^a]

inte. (x^9 - x^5)/(x^2-x^4)dx                       [e^logx =x]

int.  -x^5(x^4-1)/(x^2-1)dx

int.  -x^3(x^2+1)(x^2-1)/(x^2-1)dx

int. - x^5dx - int. x^3

-x^6/6  - x^4/4  +c

rate if useful

firstly use the 2 properties

 1 .   e^{log _e X} = X  

  2.   2 log X  = log X ²

using these properties we can modify the integral as

=  X⁹ - X⁵ /  X² - X⁴

take   X² out from numerator and denominator and cancell them off now we get

= X⁷ - X³ / 1 - X²

now take   -X³ common from numerator we get

= -X³( 1 - X⁴ ) / 1 - X²

now write 1 - X⁴ as ( 1 - X² )( 1 + X² )

( 1 - X² ) cancells with the denominator

we get

= - X³ - X⁵

= - X⁴/ 4  -  X⁶/ 6 + C