antegral(((e^5logx-e^4logx)/(e^3logx-e^2logx))dx)=
logx =v then dx=dv*logx
antegral(((e^5logx-e^4logx)/(e^3logx-e^2logx))dx)=
antegral((e^5v-e^4v)/((e^3v-e^2v)v) dv)=
1-antegral((e^5v/(e^3v-e^2v))-(e^4v/(e^3v-e^2v))dx)=
then:
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antegral((e^5v/(e^3v-e^2v)v)dv)-antegral((e^4v/(e^3v-e^2v)v)dx)
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part 1:
antegral((e^5v/(e^3v-e^2v)v)dv)
e^v=u then e^v *dv=du then du=u dv then dv=du/u
v=logu
antegral((e^5v/(e^3v-e^2v)v)dv)=ant((u^5/(u^3-u^2)vu)du)=ant((u^4/(u^3-u^2)logu)du)=

u^4/(u^3-u^2)=u^4/u^2(u-1)=u^2/(u-1)logu
u-1=p then u=p+1 then dp=du
logu=log(p+1)
ant(((u^2)/(u-1)logu)du)=ant((((p+1)^2)/(p*log(p+1)))dp)=ant(((p^2+2p+1)/p*log(p+1))dp)=
ant(((p^2)/p*log(p+1))dp)+ant((2p/p*log(p+1))dp)+ant((1/p*log(p+1))dp)=
ant((p/log(p+1))dp)+ant((2/log(p+1)dp)+ant((1/p*log(p+1))dp)=
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1.ant((p/log(p+1))dp)=

2.ant((2/log(p+1)dp)=ant((2+log(p+1)-log(p+1)/log(p+1))dp)=
ant(((2+log(p+1))/log(p+1))-(log(p+1)/log(p+1))dp)= antegral(((2+log(p+1))/log(p+1))dp)-antegral(log(p+1)/log(p+1))dp) log(p+1)=u then du=1/(p+1) dp ant(2+u/u)
3.ant((1/p*log(p+1))dp)

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then (p=u-1 ,u=e^v,p=e^v-1,v=logx,p=e^logx-1)
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result:
.....