\[ \int \frac {\text {ArcTan}(d+e x)}{a+b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {\arctan \! \left (e x +d \right ) \ln \! \left (\frac {2 e \left (b +2 c x -\sqrt {-4 a c +b^{2}}\right )}{\left (1-\mathrm {I} \left (e x +d \right )\right ) \left (2 c \left (\mathrm {I}-d \right )+e \left (b -\sqrt {-4 a c +b^{2}}\right )\right )}\right )}{\sqrt {-4 a c +b^{2}}}-\frac {\arctan \! \left (e x +d \right ) \ln \! \left (\frac {2 e \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (1-\mathrm {I} \left (e x +d \right )\right ) \left (2 c \left (\mathrm {I}-d \right )+e \left (b +\sqrt {-4 a c +b^{2}}\right )\right )}\right )}{\sqrt {-4 a c +b^{2}}}-\frac {\mathrm {I} \polylog \! \left (2, 1+\frac {4 c d -4 c \left (e x +d \right )-2 e \left (b -\sqrt {-4 a c +b^{2}}\right )}{\left (1-\mathrm {I} \left (e x +d \right )\right ) \left (2 \,\mathrm {I} c -2 c d +b e -e \sqrt {-4 a c +b^{2}}\right )}\right )}{2 \sqrt {-4 a c +b^{2}}}+\frac {\mathrm {I} \polylog \! \left (2, 1+\frac {4 c d -4 c \left (e x +d \right )-2 e \left (b +\sqrt {-4 a c +b^{2}}\right )}{\left (1-\mathrm {I} \left (e x +d \right )\right ) \left (2 c \left (\mathrm {I}-d \right )+e \left (b +\sqrt {-4 a c +b^{2}}\right )\right )}\right )}{2 \sqrt {-4 a c +b^{2}}} \]
command
Integrate[ArcTan[d + e*x]/(a + b*x + c*x^2),x]
Mathematica 13.1 output
\[ \text {\$Aborted} \]
Mathematica 12.3 output
\[ \frac {i \left (-\text {Li}_2\left (\frac {2 c (d+e x-i)}{2 c (d-i)+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )+\text {Li}_2\left (\frac {2 c (d+e x-i)}{2 c (d-i)-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )+\text {Li}_2\left (\frac {2 c (d+e x+i)}{2 c (d+i)+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )-\text {Li}_2\left (\frac {2 c (d+e x+i)}{2 c (d+i)-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )+\log (1-i (d+e x)) \log \left (\frac {e \left (\sqrt {b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c (d+i)}\right )-\log (1-i (d+e x)) \log \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt {b^2-4 a c}+b\right )-2 c (d+i)}\right )-\log (1+i (d+e x)) \log \left (\frac {e \left (\sqrt {b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c (d-i)}\right )+\log (1+i (d+e x)) \log \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt {b^2-4 a c}+b\right )-2 c (d-i)}\right )\right )}{2 \sqrt {b^2-4 a c}} \]