Evaluate: `intsqrt(tanx)`dx

Explanation

`=int sqrt(t^(2)) .(2t dt )/(1+t^(4)) " ""Let tan x = "t^(2)`
`rArr sec^(2) x dx =dt .2t`
` =int ((t^(2) +1)+(t^(2)-1))/(t^(4)+1) dt rArr dx = (2t dt)/(1+ tan^(2) x)`
`=int(t^(2) +1)/(t^(4)+1) dt + int(t^(2) -1)/(t^(4)+_1) dt " " =(2t dt)/(1+t^(4))`
` =int (1+(1)/(t^(2)))/(t^(2) +(1)/(t^(2)))dt +int(1-(1)/(t^(2)))/(t^(2)-(1)/(t^(2))) dt`
`=int (1+(1)/(t^(2)))/((t-(1)/(t))^(2) +(sqrt(2))^(2)) dt + int(1-(1)/(t^(2)))/((t+(1)/(t))^(2)-(sqrt(2))^(2)) dt`
Let for 1st intergral `t-(1)/(t)=u rArr (1+(1)/(t^(2))) dt =du`
for 2nd intergal `t+(1)/(t)=u rArr (1-(1)/(t^(2))) dt =du`
`:. I = int (1)/(u^(2) +(sqrt(2))^(2) )du +int(1)/(u^(2) -(sqrt(2))^(2)) du`
` =(1)/(sqrt(2)) tan^(-1) ((u)/(sqrt(2))) +(1)/(2sqrt(2)) log |(u-sqrt(2))/(u+sqrt(2))| +c.`
` =(1)/(sqrt(2)) tan^(-1) ((t-(1)/(t))/(sqrt(2)))+(1)/(2sqrt(2))log |(t+(1)/(t)-sqrt2)/(t+(1)/(t) +sqrt(2))| +c.`
`=(1)/(sqrt(2))tan^(-1) ((t^(2)-1)/(sqrt(2)t))+(1)/(2sqrt(2))log |(t^(2)-tsqrt2+1)/(t^(2) +tsqrt(2) +1)| +c.`
`=(1)/(sqrt(2)) tan^(-1)((tan x-1)/(sqrt(2) tan x+1))+(1)/(2sqrt(2)) log`
` |(tan x-sqrt(2 tan x+1))/(tan x+ sqrt(2tan x +1))| +c.`