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 = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.13, 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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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \sec \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (a \sec \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.46, size = 0, normalized size = 0.00 \[ \int \left (a \sec \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sec \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (\frac {a}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
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 \[ \int \left (a \sec {\left (e + f x \right )}\right )^{m} \left (b \tan {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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