Optimal. Leaf size=57 \[ \frac{\cos ^2(e+f x)^{7/3} \tan ^5(e+f x) \, _2F_1\left (\frac{7}{3},\frac{5}{2};\frac{7}{2};\sin ^2(e+f x)\right )}{5 f \sqrt [3]{d \sec (e+f x)}} \]
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Rubi [A] time = 0.0386355, 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} \[ \frac{\cos ^2(e+f x)^{7/3} \tan ^5(e+f x) \, _2F_1\left (\frac{7}{3},\frac{5}{2};\frac{7}{2};\sin ^2(e+f x)\right )}{5 f \sqrt [3]{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \frac{\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx &=\frac{\cos ^2(e+f x)^{7/3} \, _2F_1\left (\frac{7}{3},\frac{5}{2};\frac{7}{2};\sin ^2(e+f x)\right ) \tan ^5(e+f x)}{5 f \sqrt [3]{d \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.148885, size = 69, normalized size = 1.21 \[ \frac{3 \tan (e+f x) \left (9 \sqrt [3]{\cos ^2(e+f x)} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )+2 \sec ^2(e+f x)-11\right )}{16 f \sqrt [3]{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tan \left ( fx+e \right ) \right ) ^{4}{\frac{1}{\sqrt [3]{d\sec \left ( fx+e \right ) }}}}\, 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 )^{4}}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{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{2}{3}} \tan \left (f x + e\right )^{4}}{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 ^{4}{\left (e + f x \right )}}{\sqrt [3]{d \sec{\left (e + f x \right )}}}\, 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 )^{4}}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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