To solve the integral $\int \cos^{-1}(x) \sqrt{\tan(x)} \ln(\tan(x)) \, dx$, let's delve into the steps methodically. Here's the solution approach:

Given:

$\int \cos^{-1}(x) \sqrt{\tan(x)} \ln(\tan(x)) \, dx$

Let us simplify the process:

This integral is non-standard and highly complex, involving combinations of inverse trigonometric functions, roots, logarithms, and compositions of functions. Its evaluation likely requires advanced substitution techniques or numerical computation.

Key Techniques to Consider:

1. Substitution: Let $u = \cos^{-1}(x)$, then $x = \cos(u)$, and $dx = -\sin(u) du$.
2. Simplify the integral in terms of $u$, transforming all terms ($\sqrt{\tan(x)}, \ln(\tan(x))$) into expressions involving $u$.

Substitutions (sketch):

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and then write terms of $u$.
• The transformed integral may need approximation or numerical methods to compute due to its complexity.

Would you like me to break this down further or apply numerical computation for approximate evaluation?