<s>[INST] Find $\lim\limits_{t \to \infty} \int_{0}^{t} \frac{\mathrm dx}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}$ 
Find $$\lim_{t \to \infty} \int_{0}^{t} \frac{1}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}\mathrm dx$$ where $a,b,c$ are strict positive and dinstinct real numbers.

I know it should be something with arctangent but I don't know how to get there. Can somebody give me some tips, please?
 [/INST] Hints
Method 1: Partial fractions Since $a, b, c$ are positive and distinct, a straightforward application of the method of partial fractions gives that the integrand is $$\sum \frac{1}{(a^2 - b^2)(a^2 - c^2)} \frac{1}{x^2 + a^2},$$ where the sum is over the cyclic permutations of the parameters $(a, b, c)$. By linearity we can write the integral as a linear combination of integrals of the form $$\int_0^{\infty} \frac{dx}{x^2 + k^2} ,$$ and like you say, these can be evaluated with the usual antiderivative formula $$\int \frac{dx}{x^2 + k^2} = \frac{1}{k} \arctan \frac{x}{k} + C .$$
Method 2: Contour integration Since the integrand in even, it coincides with $$\frac{1}{2} \int_{-\infty}^\infty \frac{dx}{(x^2 + a^2) (x^2 + b^2) (x^2 + c^2)}.$$ This suggests integrating over a semicircle $\Gamma_R$ of radius $R$ with diameter on the real axis and centered at zero, say, in the upper half-plane. For large $R$, the value of the integral over the circular arc is $O(R^{-5})$, so in the limit its contribution to the integral is zero and hence $$\frac{1}{2} \int_{-\infty}^\infty \frac{dx}{(x^2 + a^2) (x^2 + b^2) (x^2 + c^2)} = \lim_{R \to \infty} \frac{1}{2} \oint_{\Gamma_R} \frac{dz}{(z^2 + a^2) (z^2 + b^2) (z^2 + c^2)} ,$$ but (for large $R$, and since $a, b, c$ are positive and distinct) the poles inside $\Gamma_R$ are simple poles at $ia, ib, ic$, so we can apply the residue formula, which gives that the integral is $$\frac{1}{2} \cdot 2 \pi i (\operatorname{Res}(f, ia) + \operatorname{Res}(f, ib) + \operatorname{Res}(f, ic)),$$ where $f$ is the integrand.
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