Final Answer
$\frac{1}{2}x^2+x+8\ln\left(x-1\right)-\frac{7}{3}\ln\left(x^{2}+1\right)+\frac{13}{3}\arctan\left(x\right)+\frac{-1}{x-1}+C_0$
Got another answer? Verify it here!
Step-by-step Solution
Specify the solving method
1
Divide $x^5-x^4-3x+5$ by $x^4-2x^3+2x^2-2x+1$
$\begin{array}{l}\phantom{\phantom{;}x^{4}-2x^{3}+2x^{2}-2x\phantom{;}+1;}{\phantom{;}x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;}x^{4}-2x^{3}+2x^{2}-2x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{5}-x^{4}\phantom{-;x^n}\phantom{-;x^n}-3x\phantom{;}+5\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{4}-2x^{3}+2x^{2}-2x\phantom{;}+1;}\underline{-x^{5}+2x^{4}-2x^{3}+2x^{2}-x\phantom{;}\phantom{-;x^n}}\\\phantom{-x^{5}+2x^{4}-2x^{3}+2x^{2}-x\phantom{;};}\phantom{;}x^{4}-2x^{3}+2x^{2}-4x\phantom{;}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{4}-2x^{3}+2x^{2}-2x\phantom{;}+1-;x^n;}\underline{-x^{4}+2x^{3}-2x^{2}+2x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;-x^{4}+2x^{3}-2x^{2}+2x\phantom{;}-1\phantom{;}\phantom{;}-;x^n;}-2x\phantom{;}+4\phantom{;}\phantom{;}\\\end{array}$
2
Resulting polynomial
$\int\left(x+1+\frac{-2x+4}{x^4-2x^3+2x^2-2x+1}\right)dx$
3
Expand the integral $\int\left(x+1+\frac{-2x+4}{x^4-2x^3+2x^2-2x+1}\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
$\int xdx+\int1dx+\int\frac{-2x+4}{x^4-2x^3+2x^2-2x+1}dx$
4
The integral $\int xdx$ results in: $\frac{1}{2}x^2$
$\frac{1}{2}x^2$
5
The integral $\int1dx$ results in: $x$
$x$
6
The integral $\int\frac{-2x+4}{x^4-2x^3+2x^2-2x+1}dx$ results in: $\frac{-1}{x-1}-\frac{7}{3}\ln\left(x^{2}+1\right)+\frac{13}{3}\arctan\left(x\right)+8\ln\left(x-1\right)$
$\frac{-1}{x-1}-\frac{7}{3}\ln\left(x^{2}+1\right)+\frac{13}{3}\arctan\left(x\right)+8\ln\left(x-1\right)$
7
Gather the results of all integrals
$\frac{1}{2}x^2+x+8\ln\left(x-1\right)-\frac{7}{3}\ln\left(x^{2}+1\right)+\frac{13}{3}\arctan\left(x\right)+\frac{-1}{x-1}$
8
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{1}{2}x^2+x+8\ln\left(x-1\right)-\frac{7}{3}\ln\left(x^{2}+1\right)+\frac{13}{3}\arctan\left(x\right)+\frac{-1}{x-1}+C_0$
Final Answer
$\frac{1}{2}x^2+x+8\ln\left(x-1\right)-\frac{7}{3}\ln\left(x^{2}+1\right)+\frac{13}{3}\arctan\left(x\right)+\frac{-1}{x-1}+C_0$