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How to solve the following indefinite integral? $$ \int \frac{\lambda \alpha t^\alpha}{(1+\lambda t^{\alpha})^2}dt $$ where $\lambda,\alpha$ are real numbers.

I've tried to integration by part, however I came across $\int \frac{1}{1+\lambda t^\alpha}dt$, which I still couldn't solve.

So, can anybody give me a hint? Big thanks!

sss1031
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    I didn't compute it but according to WolframAlpha there doesn't seem to be a nice closed solution but this kind of integral is expressed as the Hypergeometric Function $_2 F_1$, see http://mathworld.wolfram.com/HypergeometricFunction.html . Not really a satisfying answer but to be exact $$\int \frac{1}{1+λ t^α} dt = t_2 F_1 (1, \frac{1}{α}; 1+\frac{1}{α}; -t^α \lambda) + \text{constant}$$ see http://www.wolframalpha.com/input/?i=%5Cint+%5Cfrac%7B1%7D%7B1%2B%5Clambda+t%5E%5Calpha%7Ddt – Cahn Sep 28 '16 at 09:02
  • Thanks a lot...It turns out that I have to R or Matlab to obtain a numerical outcome. Your answer is very helpful. – sss1031 Oct 04 '16 at 07:22

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