Optimal. Leaf size=64 \[ \frac{2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{5/2} \, _2F_1\left (\frac{7}{12},\frac{5}{4};\frac{9}{4};\sin ^2(e+f x)\right )}{5 d f (b \sec (e+f x))^{4/3}} \]
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Rubi [A] time = 0.0597472, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2617} \[ \frac{2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{5/2} \, _2F_1\left (\frac{7}{12},\frac{5}{4};\frac{9}{4};\sin ^2(e+f x)\right )}{5 d f (b \sec (e+f x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \frac{(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx &=\frac{2 \cos ^2(e+f x)^{7/12} \, _2F_1\left (\frac{7}{12},\frac{5}{4};\frac{9}{4};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{5/2}}{5 d f (b \sec (e+f x))^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0768728, size = 71, normalized size = 1.11 \[ \frac{3 \left (-\tan ^2(e+f x)\right )^{3/4} \cot ^3(e+f x) (d \tan (e+f x))^{3/2} \, _2F_1\left (-\frac{2}{3},-\frac{1}{4};\frac{1}{3};\sec ^2(e+f x)\right )}{4 f (b \sec (e+f x))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.186, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( b\sec \left ( fx+e \right ) \right ) ^{-{\frac{4}{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{\left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac{4}{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 (b \sec \left (f x + e\right )\right )^{\frac{2}{3}} \sqrt{d \tan \left (f x + e\right )} d \tan \left (f x + e\right )}{b^{2} \sec \left (f x + e\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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