Need help with the integral $\int \frac{2\tan(x)+3}{5\sin^2(x)+4}\,dx$ I'm having a problem resolving the following integral, spent almost all day trying.
Any help would be appreciated.
$$\int \frac{2\tan(x)+3}{5\sin^2(x)+4}\,dx$$

$$
\begin{aligned}
& \int \frac{2 \tan x+3}{5 \sin ^{2} x+4} d x \\
=& \int \frac{2 \sec ^{2} x \tan x+3 \sec ^{2} x}{5 \tan ^{2} x+4 \sec ^{2} x} d x \\
=& 2 \int \frac{\sec ^{2} x \tan x}{5 \tan ^{2} x+4 \sec ^{2} x} d x+3 \int \frac{\sec ^{2} x}{5 \tan ^{2} x+4 \sec ^{2} x} d x \\
=& \int \frac{2sd s}{9 s^{2}-5}+\int \frac{3d t}{9 t^{2}+4} \quad \textrm{ where }s=\sec x \textrm{ and }t=\tan x.\\
=& \ln \left|9 s^{2}-5\right|+\frac{1}{2} \arctan\left(\frac{3 t}{2}\right)+C  \\
=& \ln \left|9 \sec ^{2} x-5\right|+\frac{1}{2} \arctan \left(\frac{3 \tan x}{2}\right)+C
\end{aligned}
$$
