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}