\[ \int (e x)^m \text {Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {{\mathrm e}^{\frac {\left (1+m \right ) \left (-4 a b \,d^{2} n +m +1\right )}{4 b^{2} d^{2} n^{2}}} x \left (e x \right )^{m} \erf \! \left (\frac {1+m -2 a b \,d^{2} n -2 b^{2} d^{2} n \ln \left (c \,x^{n}\right )}{2 b d n}\right ) \left (c \,x^{n}\right )^{-\frac {1+m}{n}}}{1+m}+\frac {\left (e x \right )^{1+m} \mathrm {erfc}\! \left (d \left (a +b \ln \! \left (c \,x^{n}\right )\right )\right )}{e \left (1+m \right )} \]
command
integrate((e*x)^m*erfc(d*(a+b*log(c*x^n))),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {x x^{m} \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) e^{m}}{m + 1} + \frac {x x^{m} e^{m}}{m + 1} - \frac {\pi \operatorname {erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac {m}{2 \, b d n} + \frac {1}{2 \, b d n}\right ) e^{\left (m - \frac {a m}{b n} - \frac {a}{b n} + \frac {m^{2}}{4 \, b^{2} d^{2} n^{2}} + \frac {m}{2 \, b^{2} d^{2} n^{2}} + \frac {1}{4 \, b^{2} d^{2} n^{2}}\right )}}{{\left (\pi + \pi m\right )} c^{\frac {m}{n}} c^{\left (\frac {1}{n}\right )}} \]