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