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