Paradox in indefinite integral Today,I met a problem about calculating such an indefinite integral$$ I=\int \frac{1}{(1+x^2)(1+x^{2019})}$$ My thought is as follows $$
\begin{aligned} I & = \int \frac { 1 } { \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 2019} \right) } d x \\ & = \int \frac { 1 } { \left( 1 + x ^ { - 2 } \right) \left( 1 + x ^ { - 2019} \right) } \left( - \frac { 1 } { x ^ { 2 } } \right) d x \\ & = \int \frac { - x ^ { 2019 } } { \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 2019} \right) } d x \\ & = I - \int \frac { 1 } { 1 + x ^ { 2 } } d x \end{aligned}
$$ No doubt,it is wrong.However, I don’t know how to illustrate this paradox?

In going from the first line to the second line, you make a substitution.  Thus, if
$$
I =\int \frac{1}{(1+x^2)(1+x^{2019})}dx = \int f(x)\,dx,
$$
Then
$$
\int \frac { 1 } { \left( 1 + x ^ { - 2 } \right) \left( 1 + x ^ { - 2019} \right) } \left( - \frac { 1 } { x ^ { 2 } } \right) d x = \int f(1/x)\,dx.
$$
We have no reason to believe that the two resulting antiderivatives should be identical.
