Optimal. Leaf size=40 \[ \frac {i a (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n} \]
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Rubi [A] time = 0.06, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {3493} \[ \frac {i a (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n} \]
Antiderivative was successfully verified.
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Rule 3493
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n} \, dx &=\frac {i a (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n}}{f n}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 40, normalized size = 1.00 \[ \frac {i a (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 115, normalized size = 2.88 \[ \frac {\left (\frac {2 \, d e^{\left (i \, f x + i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{2 \, n} {\left (i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-i \, e n + {\left (-i \, f n + i \, f\right )} x - 2 i \, f x - {\left (n - 1\right )} \log \left (\frac {2 \, d e^{\left (i \, f x + i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right ) - {\left (n - 1\right )} \log \left (\frac {a}{d}\right ) - i \, e\right )}}{2 \, f n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x + e\right )\right )^{2 \, n} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{-n + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.66, size = 1291, normalized size = 32.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 137, normalized size = 3.42 \[ \frac {i \, a^{-n + 1} d^{2 \, n} e^{\left (-n \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - n \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right ) - n \log \left (-\frac {2 i \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right ) + 2 \, n \log \left (-\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right )\right )}}{f n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.59, size = 62, normalized size = 1.55 \[ \frac {a\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2\,n}\,1{}\mathrm {i}}{f\,n\,{\left (\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (e+f\,x\right )}^2}\right )}^n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{2 n} \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{1 - n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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