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(x^3+x+1)/(x^2-1)...

`(x^3+x+1)/(x^2-1)`

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`(x^(3)+x+1)/(x^(2)-1) =x +(2x+1)/(x%^(2)-1)`
`:. int(x^(3)+x+1)/(x^(2)-1) dx=int x dx +int(2x+1)/(x^(2)-1)dx`
`= int x dx + int (2x+1)/((x+1)(x-1))dx`
`" Let " (2x+1)/((x+1)(x-1))=(A)/((x+1))+(B)/((x-1))`
` rArr (2x+1)/((x+1)(x-1)) +(A(x-1)+B(x+1))/((x+1)(x-1))`
`rArr 2x+1 =A (x-1) +B(x+1)`
`x= 1 " then " 2+1 =0 +B (1+1) rArr B=(3)/(2)`
`x=- 1 " then " -2+1+A (-1-1) +0 rArr A=(1)/(2)`
`:. int (x^(2) +x+1)/(x^(2)-1)dx`
`=int x dx + int (A)/((x+1)) dx+int(B)/((x-1))dx`
`= int x dx +(1)/(2) int(1)/((x+1))dx+(3)/(2) int(1)/((x+1))dx`
`=(x^(2))/(2)+(1)/(2) log |x+1|+(3)/(2)log |x-1|+C`
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