Choose among inner derivatives, completed squares and parts

Insert the factor 2 from the denominator derivative: $\frac12\int\dfrac{2e^{2x}}{e^{2x}-3}dx=\frac12\ln|e^{2x}-3|+C$.

The numerator is the derivative of $x^3+2x-8$. Integrating $(3x^2+2)(x^3+2x-8)^{-1/2}$ gives $2\sqrt{x^3+2x-8}+C$ on the real domain with positive radicand.

Complete the square: $x^2+2x+5=(x+1)^2+2^2$. The integral is $\frac12\tan^{-1}((x+1)/2)+C$.

Integrate by parts with $u=\ln x$, $dv=x^2dx$. Then $v=x^3/3$ and the answer is $\frac13x^3\ln x-\frac13\int x^2dx=\frac13x^3\ln x-\frac19x^3+C$.

Insert the inner derivative 2x: $\frac12\int(2x)\sec(x^2)\tan(x^2)dx=\frac12\sec(x^2)+C$.