Integrand size = 30, antiderivative size = 63 \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=\frac {i \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1-i \tan (e+f x))\right ) (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n}}{2 f n} \]
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Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3573, 3562, 70} \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=\frac {i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {1}{2} (1-i \tan (e+f x))\right )}{2 f n} \]
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Rule 70
Rule 3562
Rule 3573
Rubi steps \begin{align*} \text {integral}& = \left ((d \sec (e+f x))^{2 n} (a-i a \tan (e+f x))^{-n} (a+i a \tan (e+f x))^{-n}\right ) \int (a-i a \tan (e+f x))^n \, dx \\ & = \frac {\left (i a (d \sec (e+f x))^{2 n} (a-i a \tan (e+f x))^{-n} (a+i a \tan (e+f x))^{-n}\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+n}}{a-x} \, dx,x,-i a \tan (e+f x)\right )}{f} \\ & = \frac {i \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1-i \tan (e+f x))\right ) (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n}}{2 f n} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(63)=126\).
Time = 2.49 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.38 \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=-\frac {i 2^{-1+n} \left (e^{i f x}\right )^{-n} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^n \left (1+e^{2 i (e+f x)}\right ) \operatorname {Hypergeometric2F1}\left (1,1-n,2-n,1+e^{2 i (e+f x)}\right ) \sec ^{-n}(e+f x) (d \sec (e+f x))^{2 n} (\cos (f x)+i \sin (f x))^n (a+i a \tan (e+f x))^{-n}}{f (-1+n)} \]
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\[\int \left (a \left (i \tan \left (f x +e \right )+1\right )\right )^{-n} \left (d \sec \left (f x +e \right )\right )^{2 n}d x\]
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\[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{2 \, n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{n}} \,d x } \]
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\[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{2 n} \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{- n}\, dx \]
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\[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{2 \, n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{n}} \,d x } \]
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\[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{2 \, n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{n}} \,d x } \]
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Timed out. \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2\,n}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^n} \,d x \]
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