Integration of Rational Functions

If $\int \frac{a{x}^2+2bx+c}{{\left(A{x}^2+2Bx+C\right)}^2}dx, \left(\text{where} B^2\ne AC\right)$ is a rational function then which one of the following condition must be necessary$?$

If $\int \frac{x+5}{x^2+4x+5}dx=a\log \left(x^2+4x+5\right)+b{\tan }^{-1}\left(x+k\right)+$ constant, then ordered pair of $\left(a,b,k\right)$ is equal to

If the integration of the function $\frac{1}{9x^2-12x+8}$ is $\frac{1}{\mathrm{A}}{\tan }^{-1}\frac{(3x-{\displaystyle 2})}{{\displaystyle \mathrm{B}}}+C$ then $\mathrm{A}+\mathrm{B}$ is

Evaluate: $\int \frac{dx}{x^2+4x+8}$

Integrate the following functions with respect to $x$

$\frac{x+2}{\sqrt{x^2-2x+4}}$

Integrate with respect to $x$:

$\frac{1}{\sqrt{5x-6-x^2}}$

Integrate with respect to $x$:

$\frac{x-3}{x^2+2x-4}$

If $\int \frac{e^{x-1}}{\left(x^2-5x+4\right)}2xdx=A·F\left(x-1\right)+B·F\left(x-4\right)+C$ and $F\left(x\right)=\int \frac{e^x}{x}dx,$ then $\left(A, B\right)$ ordered set is

If ${\int }_0^1\frac{1}{\sqrt{3+2x-x^2}}dx=\frac{\mathrm{\pi }}{k}$, then find the value of $k$.

Evaluate the following:

$\int \frac{2x+3}{2x^2+3x-1}dx$

Integrate the following function w.r.t $x$

$\frac{{\sec }^{2}x}{2{\tan }^{2}x+7\tan x+13}$

Integrate the following function w.r.t $x$:

$\frac{\cos x}{3{\sin }^{2}x-4\sin x+1}$

Integrate the following function w.r.t $x$:

$\frac{1}{x\left[6(\log x{)}^2+7(\log x)+2\right]}$

Integrate the following function w.r.t $x$:

$\frac{x}{x^4+x^2+1}$

Integrate the following function w.r.t $x$:

$\frac{x}{4x^4-2x^2-3}$

Integrate the following function w.r.t $x$:

$\frac{1}{3-10x-25x^2}$

Integrate the following function w.r.t .$x$:

$\frac{1}{9x^2+6x+10}$

Evaluate the following:

$\int \frac{1}{a^2-b^2{x}^2}dx$

Evaluate the following:

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