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Formula for roots of trinomial $\displaystyle z^m-az^n-1$ with definite integration from paper Лахтинъ, “Выраженiе корней трехчленнаго алгебраическаго уравненiя посредствомъ опредѣленныхъ интеграловъ” (1890):

$\displaystyle z_j=e^{2j\pi i/m}+\frac{1}{2\pi i}\left(e^{(2j+1)\pi i/m}\int_0^\infty log\left(1+a\frac{t^n}{1+t^m}e^{(2j+1)\pi in/m}\right)dt \\- e^{(2j-1)\pi i/m}\int_0^\infty log\left(1+a\frac{t^n}{1+t^m}e^{(2j-1)\pi in/m}\right)dt\right)$

where natural $m>n>0$, $j=0,1,...m-1$ and $a$ is natural.

Formula work perfect for any $a,m$ and even $n$, but for odd $n$ there is an error in two roots.

Example code:

a = 7; m = 5; n = 1;
Print["\nEquation: z^", m, " - ", a, "*z^", n, " - 1 = 0\n"];
Print["Ordinary solution:"];
Print[z /. (z^m - a z^n - 1 // NSolve), "\n"];
Print["Solution with definite integration:"];
S = Table[
   Exp[2 j Pi I/m] + 
    1/(2 Pi I) (Exp[(2 j + 1) Pi I/m]*
        NIntegrate[ 
         Log[1 + a t^n/(1 + t^m) Exp[(2 j + 1) Pi I n/m]], {t, 0, 
          Infinity}] - 
       Exp[(2 j - 1) Pi I/m]*
        NIntegrate[ 
         Log[1 + a t^n/(1 + t^m) Exp[(2 j - 1) Pi I n/m]], {t, 0, 
          Infinity}]),
   {j, 0, m - 1}
   ];
Print[S];

Output:

Equation: z^5 - 7*z^1 - 1 = 0

Ordinary solution:

{-1.58871,-0.142866,0.0355442 -1.62852 I,0.0355442 +1.62852 I,1.66049}

Solution with definite integration:

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {1.58596}. NIntegrate obtained -0.71867+4.5404 I and 0.009643094017453568` for the integral and error estimates.
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {1.58596}. NIntegrate obtained -0.71867+4.5404 I and 0.009643094017453568` for the integral and error estimates.

{1.66049 +0. I,0.0355442 +1.62852 I,-1.58841+0.0000544415 I,-0.143161-0.0000544415 I,0.0355442 -1.62852 I}

If it is inaccuracy of numerical integration, then which parameters of NIntegrate need use for more correct integration?

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From the code / computations below it seems that you just need higher values for MaxRecursion. Then only slow convergence messages are given ("NIntegrate::slwcon".)

a = 7; m = 5; n = 1;
Print["\nEquation: z^", m, " - ", a, "*z^", n, " - 1 = 0\n"];
Print["Ordinary solution:"];
NSolve[(z^m - a z^n - 1)]
sol = z /. NSolve[(z^m - a z^n - 1)]

(* During evaluation of In[88]:= 
Equation: z^5 - 7*z^1 - 1 = 0


During evaluation of In[88]:= Ordinary solution: *)

(* {{z -> -1.58871}, {z -> -0.142866}, {z -> 
   0.0355442 - 1.62852 I}, {z -> 0.0355442 + 1.62852 I}, {z -> 
   1.66049}}

{-1.58871, -0.142866, 0.0355442 - 1.62852 I, 
 0.0355442 + 1.62852 I, 1.66049} *)

Print["Solution with definite integration:"]; S = 
Table[Exp[2 j Pi I/m] + 
   1/(2 Pi I) (Exp[(2 j + 1) Pi I/m]*
       NIntegrate[
        Log[1 + a t^n/(1 + t^m) Exp[(2 j + 1) Pi I n/m]], {t, 0, 
         Infinity}, MaxRecursion -> 200] - 
      Exp[(2 j - 1) Pi I/m]*
       NIntegrate[
        Log[1 + a t^n/(1 + t^m) Exp[(2 j - 1) Pi I n/m]], {t, 0, 
         Infinity}, MaxRecursion -> 200]), {j, 0, m - 1}];

(*
During evaluation of In[93]:= Solution with definite integration:

During evaluation of In[93]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

During evaluation of In[93]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. *)

S 

(* {1.66049 + 0. I, 
     0.0355442 + 1.62852 I, -1.58871 + 7.8*10^-8 I, -0.142866 - 
     7.8*10^-8 I, 0.0355442 - 1.62852 I} *)

Here we see that you have all the values in your "ordinary solution" (using a certain tolerance):

Complement[S, sol, 
 SameTest -> (Abs[#1 - #2]/Norm[{#1, #2}, Infinity] < 10^-6 &)]

(* {} *)
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