Awkward spacing for integral in exponent
Some alternatives:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
e^{G(t)} &= e^{\int\frac{t}{1+t^2}dt} \\ % original
e^{G(t)} &= e^{\int\!\frac{t}{1+t^2}\,dt} \\
e^{G(t)} &= \exp\Bigl(\int\frac{t}{1+t^2}\,dt\Bigr) \\
e^{G(t)} &= \exp\bigl({\textstyle\int\frac{t}{1+t^2}\,dt}\bigr) \\
&= e^{\frac{1}{2}\ln(1+t^2)} \\
&= \sqrt{1 + t^2}
\end{align*}
\end{document}
Does this befit you?
\documentclass{article}
\usepackage{mathtools, graphicx}
\begin{document}
\begin{align*}
e^{G(t)} &= e^{\int{\mkern-7mu\frac{t}{1+t^{\scalebox{0.8}{$ \scriptscriptstyle2 $}}} dt}} \\
&= e^{\frac{1}{2}\ln(1+t^2)} \\
&= \sqrt{1 + t^2}
\end{align*}
\end{document}