Integral of tg(x)*ln*cos(x) dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)} \tan{\left(x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$.

   To find $v{\left(x \right)}$:

   1. The integral of sine is negative cosine:

      $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

   Now evaluate the sub-integral.

2. The integral of a constant times a function is the constant times the integral of the function:

   $$\int \left(- \frac{\cos{\left(x \right)}}{x}\right)\, dx = - \int \frac{\cos{\left(x \right)}}{x}\, dx$$

   CiRule(a=1, b=0, context=cos(x)/x, symbol=x)

   So, the result is: $- \operatorname{Ci}{\left(x \right)}$

3. Add the constant of integration:

   $$- \log{\left(x \right)} \cos{\left(x \right)} + \operatorname{Ci}{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- \log{\left(x \right)} \cos{\left(x \right)} + \operatorname{Ci}{\left(x \right)}+ \mathrm{constant}$$