Optimal. Leaf size=57 \[ \frac{\cos ^2(e+f x)^{7/6} \tan ^3(e+f x) \, _2F_1\left (\frac{7}{6},\frac{3}{2};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f (d \sec (e+f x))^{2/3}} \]
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Rubi [A] time = 0.0444188, antiderivative size = 57, 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} \[ \frac{\cos ^2(e+f x)^{7/6} \tan ^3(e+f x) \, _2F_1\left (\frac{7}{6},\frac{3}{2};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f (d \sec (e+f x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \frac{\tan ^2(e+f x)}{(d \sec (e+f x))^{2/3}} \, dx &=\frac{\cos ^2(e+f x)^{7/6} \, _2F_1\left (\frac{7}{6},\frac{3}{2};\frac{5}{2};\sin ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f (d \sec (e+f x))^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.194593, size = 79, normalized size = 1.39 \[ \frac{3 \cos ^2(e+f x)^{5/6} \tan (e+f x)-\frac{3}{2} \sin (2 (e+f x)) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )}{f \cos ^2(e+f x)^{5/6} (d \sec (e+f x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \left ( d\sec \left ( fx+e \right ) \right ) ^{-{\frac{2}{3}}}}\, 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 \frac{\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}}}\,{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 (\frac{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}} \tan \left (f x + e\right )^{2}}{d \sec \left (f x + e\right )}, 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 \frac{\tan ^{2}{\left (e + f x \right )}}{\left (d \sec{\left (e + f x \right )}\right )^{\frac{2}{3}}}\, 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 \frac{\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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