Optimal. Leaf size=95 \[ \frac{\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+2 p+1)} \text{Hypergeometric2F1}\left (\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1),\frac{1}{2} (2 p+3),\sin ^2(e+f x)\right )}{f (2 p+1)} \]
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Rubi [A] time = 0.0981185, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3658, 2617} \[ \frac{\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+2 p+1)} \, _2F_1\left (\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1);\frac{1}{2} (2 p+3);\sin ^2(e+f x)\right )}{f (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 2617
Rubi steps
\begin{align*} \int (d \sec (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx &=\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)} \, _2F_1\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*}
Mathematica [A] time = 0.18626, size = 81, normalized size = 0.85 \[ \frac{\cot (e+f x) \left (-\tan ^2(e+f x)\right )^{\frac{1}{2}-p} \left (b \tan ^2(e+f x)\right )^p (d \sec (e+f x))^m \text{Hypergeometric2F1}\left (\frac{m}{2},\frac{1}{2}-p,\frac{m+2}{2},\sec ^2(e+f x)\right )}{f m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.808, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{m} \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, 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 (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m}\,{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 (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m}, 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 (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \sec{\left (e + f x \right )}\right )^{m}\, 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 (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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