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Evaluate \( \int_{0}^{\infty} \frac{\tan ^{-1}(a x) d x}{x\left(1+x^{2}\right)} \)

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in Definite Integrals by (45 points)
Evaluate \( \int_{0}^{\infty} \frac{\tan ^{-1}(a x) d x}{x\left(1+x^{2}\right)} \)
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