Optimal. Leaf size=62 \[ -\frac{3 \sqrt [12]{\cos ^2(e+f x)} \sqrt{b \sec (e+f x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{12};\frac{5}{6};\sin ^2(e+f x)\right )}{d f \sqrt [3]{d \tan (e+f x)}} \]
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Rubi [A] time = 0.0527477, antiderivative size = 62, 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} \[ -\frac{3 \sqrt [12]{\cos ^2(e+f x)} \sqrt{b \sec (e+f x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{12};\frac{5}{6};\sin ^2(e+f x)\right )}{d f \sqrt [3]{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sec (e+f x)}}{(d \tan (e+f x))^{4/3}} \, dx &=-\frac{3 \sqrt [12]{\cos ^2(e+f x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{12};\frac{5}{6};\sin ^2(e+f x)\right ) \sqrt{b \sec (e+f x)}}{d f \sqrt [3]{d \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.228715, size = 62, normalized size = 1. \[ \frac{2 d \left (-\tan ^2(e+f x)\right )^{7/6} \sqrt{b \sec (e+f x)} \, _2F_1\left (\frac{1}{4},\frac{7}{6};\frac{5}{4};\sec ^2(e+f x)\right )}{f (d \tan (e+f x))^{7/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{b\sec \left ( fx+e \right ) } \left ( d\tan \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{\sqrt{b \sec \left (f x + e\right )}}{\left (d \tan \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{\sqrt{b \sec \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac{2}{3}}}{d^{2} \tan \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{\sqrt{b \sec \left (f x + e\right )}}{\left (d \tan \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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