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\int\frac{x^2+3}{\left(x+1\right)\left(x^2+1\right)}dx

Integral of (x^2+3)/((x+1)(x^2+1))

Answer

$2\ln\left|1+x\right|+3arctan\left(x\right)-\frac{3}{2}\ln\left|1+x^2\right|+C_0$

Step-by-step explanation

Problem

$\int\frac{x^2+3}{\left(x+1\right)\left(x^2+1\right)}dx$
1

Using partial fraction decomposition, the fraction $\frac{3+x^2}{\left(1+x^2\right)\left(1+x\right)}$ can be rewritten as

$\frac{3+x^2}{\left(1+x^2\right)\left(1+x\right)}=\frac{A+Bx}{1+x^2}+\frac{C}{1+x}$

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Answer

$2\ln\left|1+x\right|+3arctan\left(x\right)-\frac{3}{2}\ln\left|1+x^2\right|+C_0$
$\int\frac{x^2+3}{\left(x+1\right)\left(x^2+1\right)}dx$

Main topic:

Integrals by partial fraction expansion

Used formulas:

6. See formulas

Time to solve it:

~ 0.9 seconds