\[ \int (e \sec (c+d x))^{-n} (a+i a \tan (c+d x))^n \, dx \]
Optimal antiderivative \[ -\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{n} \left (e \sec \left (d x +c \right )\right )^{-n}}{d n} \]
command
integrate((a+I*a*tan(d*x+c))**n/((e*sec(d*x+c))**n),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} x & \text {for}\: n = 0 \wedge \left (d = 0 \vee n = 0\right ) \\x \left (e \sec {\left (c \right )}\right )^{- n} \left (i a \tan {\left (c \right )} + a\right )^{n} & \text {for}\: d = 0 \\- \frac {i \left (e \sec {\left (c + d x \right )}\right )^{- n} \left (i a \tan {\left (c + d x \right )} + a\right )^{n}}{d n} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} x & \text {for}\: n = 0 \\x \left (e \sec {\left (c \right )}\right )^{- n} \left (i a \tan {\left (c \right )} + a\right )^{n} & \text {for}\: d = 0 \\\int \left (0^{\frac {1}{n}} \sec {\left (c + d x \right )}\right )^{- n} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx & \text {for}\: e = 0^{\frac {1}{n}} \\- \frac {i e^{- n} \left (i a \tan {\left (c + d x \right )} + a\right )^{n} \sec ^{- n}{\left (c + d x \right )}}{d n} & \text {otherwise} \end {cases} \]________________________________________________________________________________________