Problem number 1319

7.3 Problem number 1319

$\int {(c (d \tan (e + f x))^{p})}^{n} (a + i a \tan (e + f x))^{2} d x$

Optimal antiderivative $- \frac{a^{2} \tan (f x + e) {(c {(d \tan (f x + e))}^{p})}^{n}}{f (n p + 1)} + \frac{2 a^{2} hypergeom ([1, n p + 1], [n p + 2], I \tan (f x + e)) \tan (f x + e) {(c {(d \tan (f x + e))}^{p})}^{n}}{f (n p + 1)}$

command

Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^2,x]

Mathematica 13.1 output

$\int {(c (d \tan (e + f x))^{p})}^{n} (a + i a \tan (e + f x))^{2} d x$

Mathematica 12.3 output

$\frac{a^{2} e^{- 2 i e} 2^{- n p} {(- \frac{i (- 1 + e^{2 i (e + f x)})}{1 + e^{2 i (e + f x)}})}^{n p + 1} (\cos (e + f x) + i \sin (e + f x))^{2} (- 2^{n p} + {(1 + e^{2 i (e + f x)})}^{n p + 1}_{2} F_{1} (n p + 1, n p + 1; n p + 2; \frac{1}{2} (1 - e^{2 i (e + f x)}))) \tan^{- n p} (e + f x) {(c (d \tan (e + f x))^{p})}^{n}}{(f n p + f) (\cos (f x) + i \sin (f x))^{2}}$

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