Optimal. Leaf size=57 \[ \frac{\cos ^2(e+f x)^{19/6} \tan ^5(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left (\frac{5}{2},\frac{19}{6};\frac{7}{2};\sin ^2(e+f x)\right )}{5 f} \]
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Rubi [A] time = 0.04203, 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} \[ \frac{\cos ^2(e+f x)^{19/6} \tan ^5(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left (\frac{5}{2},\frac{19}{6};\frac{7}{2};\sin ^2(e+f x)\right )}{5 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int (d \sec (e+f x))^{4/3} \tan ^4(e+f x) \, dx &=\frac{\cos ^2(e+f x)^{19/6} \, _2F_1\left (\frac{5}{2},\frac{19}{6};\frac{7}{2};\sin ^2(e+f x)\right ) (d \sec (e+f x))^{4/3} \tan ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 1.0984, size = 92, normalized size = 1.61 \[ \frac{3 d \sqrt [3]{d \sec (e+f x)} \left (-18 \sin (e+f x) \sqrt [6]{\cos ^2(e+f x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )+27 \sin (e+f x)+\tan (e+f x) \sec (e+f x) \left (7 \sec ^2(e+f x)-16\right )\right )}{91 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{{\frac{4}{3}}} \left ( \tan \left ( fx+e \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}} d \sec \left (f x + e\right ) \tan \left (f x + e\right )^{4}, 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 \left (d \sec \left (f x + e\right )\right )^{\frac{4}{3}} \tan \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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