Optimal. Leaf size=57 \[ \frac{\cos ^2(e+f x)^{5/3} \tan ^3(e+f x) \sqrt [3]{d \sec (e+f x)} \, _2F_1\left (\frac{3}{2},\frac{5}{3};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f} \]
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Rubi [A] time = 0.0364821, 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} \[ \frac{\cos ^2(e+f x)^{5/3} \tan ^3(e+f x) \sqrt [3]{d \sec (e+f x)} \, _2F_1\left (\frac{3}{2},\frac{5}{3};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int \sqrt [3]{d \sec (e+f x)} \tan ^2(e+f x) \, dx &=\frac{\cos ^2(e+f x)^{5/3} \, _2F_1\left (\frac{3}{2},\frac{5}{3};\frac{5}{2};\sin ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.280346, size = 80, normalized size = 1.4 \[ \frac{3 \sqrt [3]{d \sec (e+f x)} \left (2 \sqrt [3]{\cos ^2(e+f x)} \tan (e+f x)-\sin (2 (e+f x)) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};\sin ^2(e+f x)\right )\right )}{8 f \sqrt [3]{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{d\sec \left ( fx+e \right ) } \left ( \tan \left ( fx+e \right ) \right ) ^{2}\, 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 \left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}} \tan \left (f x + e\right )^{2}\,{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 (\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{d \sec{\left (e + f x \right )}} \tan ^{2}{\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 \left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}} \tan \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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