Optimal. Leaf size=82 \[ \frac{(a \sec (e+f x))^m (b \tan (e+f x))^{n+1} \cos ^2(e+f x)^{\frac{1}{2} (m+n+1)} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{n+3}{2};\sin ^2(e+f x)\right )}{b f (n+1)} \]
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Rubi [A] time = 0.045036, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2617} \[ \frac{(a \sec (e+f x))^m (b \tan (e+f x))^{n+1} \cos ^2(e+f x)^{\frac{1}{2} (m+n+1)} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{n+3}{2};\sin ^2(e+f x)\right )}{b f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int (a \sec (e+f x))^m (b \tan (e+f x))^n \, dx &=\frac{\cos ^2(e+f x)^{\frac{1}{2} (1+m+n)} \, _2F_1\left (\frac{1+n}{2},\frac{1}{2} (1+m+n);\frac{3+n}{2};\sin ^2(e+f x)\right ) (a \sec (e+f x))^m (b \tan (e+f x))^{1+n}}{b f (1+n)}\\ \end{align*}
Mathematica [A] time = 0.144983, size = 80, normalized size = 0.98 \[ \frac{b \left (-\tan ^2(e+f x)\right )^{\frac{1-n}{2}} (a \sec (e+f x))^m (b \tan (e+f x))^{n-1} \, _2F_1\left (\frac{m}{2},\frac{1-n}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.748, size = 0, normalized size = 0. \begin{align*} \int \left ( a\sec \left ( fx+e \right ) \right ) ^{m} \left ( b\tan \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sec \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \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 (a \sec \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sec{\left (e + f x \right )}\right )^{m} \left (b \tan{\left (e + f x \right )}\right )^{n}\, dx \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 (a \sec \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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