O17-C035
> restart;
> f:=n->ln(t)/product(t+k,k=1..n+1);

                                     ln(t)
                     f := n -> ------------------
                                 n + 1
                                --------'
                               '  |  |
                                  |  |    (t + k)
                                  |  |
                                  |  |
                                 k = 1

> int1:=int(f(1),t=0..infinity);

                                           2
                          int1 := 1/2 ln(2)

> with(IntegrationTools):
> V := Int(f(1),t=0..infinity);
> V1:=Change(V, t=2/u);

                         infinity
                        /
                       |               ln(t)
                 V :=  |          --------------- dt
                       |          (t + 1) (t + 2)
                      /
                        0


                         infinity
                        /
                       |             ln(2) - ln(u)
               V1 := - |          - --------------- du
                       |            (2 + u) (1 + u)
                      /
                        0

> int(ln(2)/(1+u)/(2+u)/2,u=0..infinity);

                                       2
                              1/2 ln(2)

> int2:=int(f(2),t=0..infinity);

                                     2            2
                   int2 := -1/4 ln(3)  + 1/2 ln(2)

> assume (a>0);int(ln(t)/(t+a)^2,t=0..infinity);

                                ln(a~)
                                ------
                                  a~

> J:=n->int(f(n),t=0..infinity);seq(J(n),n=1..10);

                                 infinity
                                /
                               |
                    J := n ->  |          f(n) dt
                               |
                              /
                                0


           2            2            2            2             2
  1/2 ln(2) , -1/4 ln(3)  + 1/2 ln(2) , -1/4 ln(3)  + 7/12 ln(2) ,

                  2             2             2
        -1/8 ln(3)  + 5/12 ln(2)  - 1/48 ln(5) ,

        23       2                                 2             2
        --- ln(2)  + 1/120 ln(2) ln(3) - 3/80 ln(3)  - 1/48 ln(5) ,
        120

        23       2                                  2               2
        --- ln(2)  + 1/120 ln(2) ln(3) - 1/160 ln(3)  - 1/1440 ln(7)
        360

                     2   59       2               2
         - 1/96 ln(5) , ---- ln(2)  - 1/1440 ln(7)
                        3360

                                          2              2
         + 1/240 ln(2) ln(3) - 1/288 ln(5) , 1/3360 ln(3)

                      2                                   2
         + 1/224 ln(2)  + 1/720 ln(2) ln(3) - 1/1152 ln(5)

                       2   199        2
         - 1/2880 ln(7) , ------ ln(2)  + 1/362880 ln(2) ln(5)
                          181440

             25        2               2                2
         - ------ ln(5)  - 1/8640 ln(7)  + 1/13440 ln(3)
           145152

                                                2    229        2
         + 1/2880 ln(2) ln(3), -1/7257600 ln(11)  + ------ ln(2)
                                                    907200

                                               2
         + 1/362880 ln(2) ln(5) - 1/36288 ln(5)

                                               2                2
         + 1/14400 ln(2) ln(3) + 1/268800 ln(3)  - 1/34560 ln(7)

> g:=n->ln(t)/product(t+a[k],k=1..n+1);

                                     ln(t)
                   g := n -> ---------------------
                               n + 1
                              --------'
                             '  |  |
                                |  |    (t + a[k])
                                |  |
                                |  |
                               k = 1

> assume(a[1]>0,a[2]>0);intgen1:=int(g(1),t=0..infinity);

                                        2            2
                               ln(a~[1])  - ln(a~[2])
                intgen1 := 1/2 -----------------------
                                   -a~[2] + a~[1]

> convert(1/(t+b)/(t+a),parfrac,t);

                          1                  1
                - ----------------- + ----------------
                  (a~ - b) (t + a~)   (a~ - b) (t + b)

> In:=proc(a,n)
> local alpha,b,c;
> if n=1 then 
>           return int(ln(t)/(t+a[1])/(t+a[2]),t=0..infinity)
> else
>           alpha:=1/(a[n+1]-a[n]);
>           b:=[seq(a[k],k=1..n)];
>           c:=[seq(a[k],k=1..n-1),a[n+1]];
>           return((In(b,n-1)-In(c,n-1))*alpha)
> fi;
> end;

  In := proc(a, n)
local alpha, b, c;
    if n = 1 then return
        int(ln(t)/((t + a[1])*(t + a[2])), t = 0 .. infinity)
    else
        alpha := 1/(a[n + 1] - a[n]);
        b := [seq(a[k], k = 1 .. n)];
        c := [seq(a[k], k = 1 .. n - 1), a[n + 1]];
        return (In(b, n - 1) - In(c, n - 1))*alpha
    end if
end proc

> In([a,b],1);

         {        undefined                And(a~ < 0, b < 0)
         {
         {        undefined                      a~ < 0
         {
         {        undefined                      b < 0
         {
         {         1   2          2
         {     ln(----)  - ln(1/b)
         {         a~
         { 1/2 --------------------            otherwise
         {            a~ - b

> In([1,2,3],2);

                                 2            2
                       -1/4 ln(3)  + 1/2 ln(2)

> seq(In([seq(k,k=1..n+1)],n),n=1..10);

           2            2            2            2             2
  1/2 ln(2) , -1/4 ln(3)  + 1/2 ln(2) , -1/4 ln(3)  + 7/12 ln(2) ,

                  2             2             2
        -1/8 ln(3)  + 5/12 ln(2)  - 1/48 ln(5) ,

        23       2                                 2             2
        --- ln(2)  + 1/120 ln(2) ln(3) - 3/80 ln(3)  - 1/48 ln(5) ,
        120

        23       2                                  2               2
        --- ln(2)  + 1/120 ln(2) ln(3) - 1/160 ln(3)  - 1/1440 ln(7)
        360

                     2   59       2               2
         - 1/96 ln(5) , ---- ln(2)  - 1/1440 ln(7)
                        3360

                                          2              2
         + 1/240 ln(2) ln(3) - 1/288 ln(5) , 1/3360 ln(3)

                      2                                   2
         + 1/224 ln(2)  + 1/720 ln(2) ln(3) - 1/1152 ln(5)

                       2   199        2
         - 1/2880 ln(7) , ------ ln(2)  + 1/362880 ln(2) ln(5)
                          181440

             25        2               2                2
         - ------ ln(5)  - 1/8640 ln(7)  + 1/13440 ln(3)
           145152

                                                2    229        2
         + 1/2880 ln(2) ln(3), -1/7257600 ln(11)  + ------ ln(2)
                                                    907200

                                               2
         + 1/362880 ln(2) ln(5) - 1/36288 ln(5)

                                               2                2
         + 1/14400 ln(2) ln(3) + 1/268800 ln(3)  - 1/34560 ln(7)

>