evalin(symengine,'int(log(1+x*sqrt(1+x^2)),x)')

ans =
 
x*log(x*(x^2 + 1)^(1/2) + 1) - 2*x + (log(x - (2^(1/2)*(5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 - 5/2))/(2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2)) - (log(x + (2^(1/2)*(5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 - 5/2))/(2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2)) - (log(x - (2^(1/2)*(- 5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 + 5/2))/(2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3/2)) + (log(x + (2^(1/2)*(- 5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 + 5/2))/(2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3/2)) - (log(((2^(1/2)*x*(5^(1/2) - 1)^(1/2))/2 + (2^(1/2)*(x^2 + 1)^(1/2)*(5^(1/2) + 1)^(1/2))/2 + 1)/(x - (2^(1/2)*(5^(1/2) - 1)^(1/2))/2))*((5^(1/2)/2 - 1/2)^(1/2) + 2*(5^(1/2)/2 - 1/2)^(3/2)))/((2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2))*(5^(1/2)/2 + 1/2)^(1/2)) - (log(((2^(1/2)*(x^2 + 1)^(1/2)*(5^(1/2) + 1)^(1/2))/2 - (2^(1/2)*x*(5^(1/2) - 1)^(1/2))/2 + 1)/(x + (2^(1/2)*(5^(1/2) - 1)^(1/2))/2))*((5^(1/2)/2 - 1/2)^(1/2) + 2*(5^(1/2)/2 - 1/2)^(3/2)))/((2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2))*(5^(1/2)/2 + 1/2)^(1/2)) - (log(((2^(1/2)*x*(- 5^(1/2) - 1)^(1/2))/2 + (2^(1/2)*(- x^2 - 1)^(1/2)*(5^(1/2) - 1)^(1/2))/2 + 1)/(x - (2^(1/2)*(- 5^(1/2) - 1)^(1/2))/2))*((- 5^(1/2)/2 - 1/2)^(1/2) + 2*(- 5^(1/2)/2 - 1/2)^(3/2)))/((2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3/2))*(1/2 - 5^(1/2)/2)^(1/2)) - (log(((2^(1/2)*(- x^2 - 1)^(1/2)*(5^(1/2) - 1)^(1/2))/2 - (2^(1/2)*x*(- 5^(1/2) - 1)^(1/2))/2 + 1)/(x + (2^(1/2)*(- 5^(1/2) - 1)^(1/2))/2))*((- 5^(1/2)/2 - 1/2)^(1/2) + 2*(- 5^(1/2)/2 - 1/2)^(3/2)))/((2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3/2))*(1/2 - 5^(1/2)/2)^(1/2))
 
EDU>> pretty(ans)
 
                           log(x - #5) #6   log(x + #5) #6   log(x - #4) #7   log(x + #4) #7 
  x log(x #13 + 1) - 2 x + -------------- - -------------- - -------------- + -------------- - 
                                #15              #15              #14              #14 
   
        / #3 + #8 + 1 \          / #8 - #3 + 1 \          / #2 + #1 + 1 \          / #1 - #2 + 1 \ 
     log| ----------- | #12   log| ----------- | #12   log| ----------- | #11   log| ----------- | #11 
        \   x - #5    /          \   x + #5    /          \   x - #4    /          \   x + #4    / 
     ---------------------- - ---------------------- - ---------------------- - ---------------------- 
              #10                      #10                       #9                       #9 
   
  where 
   
            1/2     2     1/2   1/2     1/2 
           2    (- x  - 1)    (5    - 1) 
     #1 == -------------------------------- 
                          2 
   
            1/2       1/2     1/2 
           2    x (- 5    - 1) 
     #2 == ---------------------- 
                     2 
   
            1/2     1/2     1/2 
           2    x (5    - 1) 
     #3 == -------------------- 
                    2 
   
            1/2     1/2     1/2 
           2    (- 5    - 1) 
     #4 == -------------------- 
                    2 
   
            1/2   1/2     1/2 
           2    (5    - 1) 
     #5 == ------------------ 
                   2 
   
            1/2 
           5 
     #6 == ---- - 5/2 
            2 
   
            1/2 
           5 
     #7 == ---- + 5/2 
            2 
   
            1/2       1/2     1/2 
           2    #13 (5    + 1) 
     #8 == ---------------------- 
                     2 
   
               /        1/2 \1/2 
               |       5    | 
     #9 == #14 | 1/2 - ---- | 
               \        2   / 
   
                /  1/2       \1/2 
                | 5          | 
     #10 == #15 | ---- + 1/2 | 
                \  2         / 
   
               1/2        3/2 
     #11 == #16    + 2 #16 
   
               1/2        3/2 
     #12 == #17    + 2 #17 
   
              2     1/2 
     #13 == (x  + 1) 
   
                 1/2        3/2 
     #14 == 2 #16    + 4 #16 
   
                 1/2        3/2 
     #15 == 2 #17    + 4 #17 
   
               1/2 
              5 
     #16 == - ---- - 1/2 
               2 
   
             1/2 
            5 
     #17 == ---- - 1/2 
             2