Does all proofs by contrapositive have a direct proof that is similar in length? I know this may be a question not suitable to ask here or is a duplicate, but I'm just wondering if all proofs by contrapositives can be rephrased into a direct proof, without using the fact that they are logically equivalent?
Are there instances when a proof by contrapositive is not too hard, but direct proof is way more involved or literally impossible to do?
 A: I assume below that by "proof by contrapositive" you mean the principle

You can prove $A\rightarrow B$ by showing that $\lnot B$ implies $\lnot A$.

or in other words

From $\lnot B\rightarrow \lnot A$ you may infer $A\rightarrow B$.

[EDIT: The above lines were added following Rob Arthan's comments]
Proof by contrapositive is a reasoning principle which is valid in classical logic, but not in intuitionistic logic. So if you have a proof by contrapositive of a theorem which is not intuitionistically valid (e.g., $\lnot\lnot A\rightarrow A$), you cannot really get rid of the contrapositive reasoning: At best (depending on the precise formulation of your proof system) you can replace it by another equivalent "non-constructive" principle such as double negation elimination or proof by excluded middle.
So how about classical theorems which are also intuitionistically valid? Here you can remove all non-constructive reasoning steps in proofs, but this might lead to much larger proofs. A nice and simple example of this is blowup is mentioned in the Introduction of:

M. Baaz, A. Leitsch and G. Reis, "A Note on the Complexity of Classical and Intuitionistic Proofs," 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, Kyoto, Japan, 2015, pp. 657-666, doi: 10.1109/LICS.2015.66.

For an intuitionistically valid theorem $A$, simply consider the formula $B:=A\lor\lnot A$. An intuitionistic proof of $B$ will essentially go through the proof of $A$ and the conclude $A\lor \lnot A$ in the last step, and so it can be as large as intuitionistic proofs can get (there is no computable upper bound on the size of intuitionistic proof of a (first-order) theorem $A$ relative to the size of $A$)! On the other hand, $B$ has a trivial classical proof as it is an instance of the law of excluded middle.
A: Some existence theorems go into that direction: First one assumes that a solution does not exist, then use a contradiction argument using some deep (and non-constructive) theorem, like for instance Brouwer fixed point theorem or Hahn-Banach theorem.
