$\int \frac{x^2+1}{x^4-x^2+1} \, dx =$

Integrate the function: $\frac{x+2}{\sqrt{x^{2}-1}}$

If $\int \frac{3e^x - 5e^{-x}}{4e^x + 5e^{-x}} dx = px + q \cdot \log |4e^x + 5e^{-x}| + C$, then

If $\int(\sin x )^{\frac{-11}{2}}(\cos x )^{\frac{-5}{2}} dx = -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+C,$ where $p_i$ and $q_i$ are positive integers with $\operatorname{gcd}(p_i, q_i)=1$ for $i =1,2,3,4$ and $C$ is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to . . . . . . .

Let $I(x)=\int\frac{3dx}{(4x+6)(\sqrt{4x^{2}+8x+3})}$ and $I(0)=\frac{\sqrt{3}}{4}+20$. If $I(\frac{1}{2})=\frac{a\sqrt{2}}{b}+c$, where $a, b, c \in N$ and $gcd(a,b)=1$, then $a+b+c$ is equal to:

Let $f(x) = 1 - \frac{1}{x}$, $g_2(x) = f(f(x))$, $g_3(x) = f(f(f(x)))$ and so on. If $\int x \cdot g_{2026}(x) \, dx = \int g_{2025}(x) \, dx + h(x) + c$, then $h(x) = $