Optimal. Leaf size=92 \[ \frac{2 i a^2 (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n (n+1)}+\frac{i a (a+i a \tan (e+f x))^{1-n} (d \sec (e+f x))^{2 n}}{f (n+1)} \]
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Rubi [A] time = 0.127938, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {3494, 3493} \[ \frac{2 i a^2 (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n (n+1)}+\frac{i a (a+i a \tan (e+f x))^{1-n} (d \sec (e+f x))^{2 n}}{f (n+1)} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{2-n} \, dx &=\frac{i a (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n}}{f (1+n)}+\frac{(2 a) \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n} \, dx}{1+n}\\ &=\frac{i a (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n}}{f (1+n)}+\frac{2 i a^2 (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n}}{f n (1+n)}\\ \end{align*}
Mathematica [A] time = 1.07498, size = 61, normalized size = 0.66 \[ -\frac{a^2 (n \tan (e+f x)-i (n+2)) (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.809, 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 ) ^{2-n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.05998, size = 410, normalized size = 4.46 \begin{align*} \frac{2^{n + 1} a^{2} d^{2 \, n} \cos \left (n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - i \cdot 2^{n + 1} a^{2} d^{2 \, n} \sin \left (n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + 2 \,{\left (a^{2} d^{2 \, n} n + a^{2} d^{2 \, n}\right )} 2^{n} \cos \left (-2 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 2 \, e\right ) -{\left (2 i \, a^{2} d^{2 \, n} n + 2 i \, a^{2} d^{2 \, n}\right )} 2^{n} \sin \left (-2 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 2 \, e\right )}{{\left (-i \, a^{n} n^{2} - i \, a^{n} n +{\left (-i \, a^{n} n^{2} - i \, a^{n} n\right )} \cos \left (2 \, f x + 2 \, e\right ) +{\left (a^{n} n^{2} + a^{n} n\right )} \sin \left (2 \, f x + 2 \, e\right )\right )}{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac{1}{2} \, n} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90475, size = 304, normalized size = 3.3 \begin{align*} \frac{{\left ({\left (i \, n + i\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (i \, n + 2 i\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} \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 + 2} \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} e^{\left (-4 i \, f x - 4 i \, e\right )}}{2 \,{\left (f n^{2} + f n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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