$\int \frac{7x^3-86x^2\+346x-455}{x^4-15x^3\+84x^2-208 x\+192} \mathit{\ⅆ}x$

$\= \int \frac{2}{x-3}\mathit{\ⅆ}x\+\int \frac{5}{x-4}\mathit{\ⅆ}x-\int \frac{7}{{\left(x-4\right)}^2}\mathit{\ⅆ}x\+\int \frac{1}{{\left(x-4\right)}^3}\mathit{\ⅆ}x$

$\=2 \ln \left(\left|x-3\right|\right)\+5 \ln \(\|x-4\|\)\+7{\left(x-4\right)}^{-1}-{\left(x-4\right)}^{-2}\/2$

Evaluation in Maple

$\int \frac{7x^3-86x^2\+346x-455}{x^4-15x^3\+84x^2-208 x\+192} \mathit{\ⅆ}x$ = $5\ln \left(x-4\right)\+\frac{7}{x-4}-\frac{1}{2{\left(x-4\right)}^2}\+2\ln \left(x-3\right)$$$

$\begin{array}{cccc}\multicolumn{4}{c}{\int \frac{7x^3-86x^2+346x-455}{x^4-15x^3+84x^2-208x+192}\mathit{\ⅆ}x}\\ & \text{=}& -7\int \frac{1}{{\left(x-4\right)}^2}\mathit{\ⅆ}x+\int \frac{1}{{\left(x-4\right)}^3}\mathit{\ⅆ}x+5\int \frac{1}{x-4}\mathit{\ⅆ}x+2\int \frac{1}{x-3}\mathit{\ⅆ}x\hfill & \left[\mathrm{partialfractions}\right]\end{array}$

Table 6.5.2(a)   Partial Fractions rule applied in Integration Methods tutor