Integrand size = 23, antiderivative size = 95 \[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+m+2 p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1+m+2 p),\frac {1}{2} (3+2 p),\sin ^2(e+f x)\right ) (d \sec (e+f x))^m \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)} \]
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Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3739, 2697} \[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (m+2 p+1)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (2 p+1),\frac {1}{2} (m+2 p+1),\frac {1}{2} (2 p+3),\sin ^2(e+f x)\right )}{f (2 p+1)} \]
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Rule 2697
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int (d \sec (e+f x))^m \tan ^{2 p}(e+f x) \, dx \\ & = \frac {\cos ^2(e+f x)^{\frac {1}{2} (1+m+2 p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1+m+2 p),\frac {1}{2} (3+2 p),\sin ^2(e+f x)\right ) (d \sec (e+f x))^m \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {m}{2},\frac {1}{2}-p,\frac {2+m}{2},\sec ^2(e+f x)\right ) (d \sec (e+f x))^m \left (-\tan ^2(e+f x)\right )^{\frac {1}{2}-p} \left (b \tan ^2(e+f x)\right )^p}{f m} \]
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\[\int \left (d \sec \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
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\[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \sec {\left (e + f x \right )}\right )^{m}\, dx \]
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\[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \]
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