I came up with this myself while messing around on Desmos and an Integral Calculator. The solutions for $I_a=\int_{0}^{\infty}\left(\frac{\pi}{2}-\arctan\left(x^{a}\right)\right)dx$ get crazy kind of fast. Here is a chart of some of the first few solutions:
$$\begin{array}{c|c} a & I_a \\ \hline 1 & \text{Diverges} \\ \hline 2 & \frac{\pi}{\sqrt{2}} \\ \hline 3 & \frac{\pi}{\sqrt{3}} \\ \hline \end{array}$$
I thought I saw a pattern emerging here but then I tried $I_4$: $$I_4=-\dfrac{\sqrt{2}\,\sqrt{\sqrt{2}+1}\,\left(1133655784868091786638060702359755612313549400848035957003851845084004266410174146539466856676323967580590611790253331717507529115346995957976122515036285\cdot {2}^{\frac{7}{4}}+1603231386023171146104033269324389258904241774973315372455752057048906486582367709365528217283061476240674162703114834989988160878689001637598599046377397\cdot 2\,\sqrt[{4}]{2}\right)\,\pi}{3870542955759354719380154674043900483531340576669387286463455747216915019402716002444461930635709411401855386283621498425003219109382993553550844076449967\cdot 8\,\sqrt{2}+43790194734260206923873503546946317939484658813141621271353662434126572047880669694479921183350167101140236391893890667319931039904575961529195544982618914}-\dfrac{3206462772046342292208066538648778517808483549946630744911504114097812973164735418731056434566122952481348325406229669979976321757378003275197198092754795\,\sqrt{\sqrt{2}+1}\,\sqrt[{4}]{8}\,\pi}{3870542955759354719380154674043900483531340576669387286463455747216915019402716002444461930635709411401855386283621498425003219109382993553550844076449967\cdot 8\,\sqrt{2}+43790194734260206923873503546946317939484658813141621271353662434126572047880669694479921183350167101140236391893890667319931039904575961529195544982618914}-\dfrac{4534623139472367146552242809439022449254197603392143828015407380336017065640696586157867426705295870322362447161013326870030116461387983831904490060145139\cdot \sqrt[{4}]{2}\,\sqrt{\sqrt{2}+1}\,\pi}{3870542955759354719380154674043900483531340576669387286463455747216915019402716002444461930635709411401855386283621498425003219109382993553550844076449967\cdot 8\,\sqrt{2}+43790194734260206923873503546946317939484658813141621271353662434126572047880669694479921183350167101140236391893890667319931039904575961529195544982618914}$$
Which was quite unexpected. I got similar results for $I_5$ and so on.
Some other nice results I found are $I_\frac{3}{2}=\pi$, $I_\frac{4}{3}=\frac{\sqrt{2+\sqrt{2}}}{2}\pi$. $I$ appears to have niceish results for rational $1<a\le3$.
I also know that $\lim\limits_{a\to\infty} I_a=\frac{\pi}{2}$
\begin{align} &\int_{0}^{\infty}\left(\frac{\pi}{2}-\arctan x^{a}\right)dx\\ = &\int_{0}^{\infty}\text{arccot}\ x^{a}\ dx \overset{ibp}=\int_0^\infty \frac {a x^{a}}{1+x^{2a}}dx =\frac{\pi}{2}\sec\frac{\pi}{2a} \end{align}