$$ \displaystyle\int \cos\left({x}^{2}\right)\, \mathrm d x = -{{\sqrt{\pi}\,\left(i\,\mathrm{erf}\left({{\left(i+1\right)\,x }\over{\sqrt{2}}}\right)-\mathrm{erf}\left({{\left(i+1\right)\,x }\over{\sqrt{2}}}\right)+i\,\mathrm{erf}\left({{\left(i-1\right)\,x }\over{\sqrt{2}}}\right)+\mathrm{erf}\left({{\left(i-1\right)\,x }\over{\sqrt{2}}}\right)-i\,\mathrm{erf}\left(\sqrt{-i}\,x\right)- \mathrm{erf}\left(\sqrt{-i}\,x\right)+i\,\mathrm{erf}\left(\left(-1 \right)^{{{1}\over{4}}}\,x\right)-\mathrm{erf}\left(\left(-1\right) ^{{{1}\over{4}}}\,x\right)\right)}\over{8\sqrt{2}}} $$

Integral Calculator