Optimal. Leaf size=66 \[ \frac{i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \text{Hypergeometric2F1}\left (2,n,n+1,\frac{1}{2} (1-i \tan (e+f x))\right )}{4 a f n} \]
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Rubi [A] time = 0.182586, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3505, 3522, 3487, 68} \[ \frac{i (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n} \text{Hypergeometric2F1}\left (2,n,n+1,\frac{1}{2} (1-i \tan (e+f x))\right )}{4 a f n} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3522
Rule 3487
Rule 68
Rubi steps
\begin{align*} \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-1-n} \, dx &=\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 \frac{(a-i a \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx\\ &=\frac{\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 \cos ^2(e+f x) (a-i a \tan (e+f x))^{1+n} \, dx}{a^2}\\ &=\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 ) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+n}}{(a-x)^2} \, dx,x,-i a \tan (e+f x)\right )}{f}\\ &=\frac{i \, _2F_1\left (2,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}}{4 a f n}\\ \end{align*}
Mathematica [B] time = 13.9642, size = 165, normalized size = 2.5 \[ \frac{i e^{i e} 2^{n-2} \left (1+e^{2 i (e+f x)}\right )^2 \left (e^{i f x}\right )^{-n} \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^n \sec ^{1-n}(e+f x) (\cos (f x)+i \sin (f x))^{n+1} \text{Hypergeometric2F1}\left (2,2-n,3-n,1+e^{2 i (e+f x)}\right ) (a+i a \tan (e+f x))^{-n-1} (d \sec (e+f x))^{2 n}}{f (n-2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.344, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{2\,n} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{-1-n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\frac{2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{-n - 1} \left (\frac{2 \, d e^{\left (i \, f x + i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{2 \, n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{2 \, n}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{-n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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