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Find the integral $\int \tan (5x) \tan (3x) \tan(2x) \ dx $ .

This question is posted in a maths group in Facebook. What way should we use to solve integral like this? Thanks in advance

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Let $t_n = \tan(nx)$, we have

$$t_5 = \frac{t_3 + t_2}{1 – t_3t_2} \iff t_5 – t_5t_3t_2 = t_3 + t_2

\implies t_5t_3 t_2 = t_5 – t_3 – t_2$$

This leads to

$$\begin{align}\int \tan(5x)\tan(3x)\tan(2x) dx

&= \int \left(\tan(5x) – \tan(3x) – \tan(2x) \right)dx\\

&= \frac12 \log\cos(2x) + \frac13\log\cos(3x) – \frac15\log\cos(5x) + \text{ const. }

\end{align}$$

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