Optimal. Leaf size=53 \[ \frac{3 \sin (e+f x) \sec ^{\frac{13}{3}}(e+f x) \, _2F_1\left (-\frac{13}{6},-\frac{3}{2};-\frac{7}{6};\cos ^2(e+f x)\right )}{13 f \sqrt{\sin ^2(e+f x)}} \]
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Rubi [A] time = 0.0594236, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2632, 2576} \[ \frac{3 \sin (e+f x) \sec ^{\frac{13}{3}}(e+f x) \, _2F_1\left (-\frac{13}{6},-\frac{3}{2};-\frac{7}{6};\cos ^2(e+f x)\right )}{13 f \sqrt{\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2632
Rule 2576
Rubi steps
\begin{align*} \int \sec ^{\frac{16}{3}}(e+f x) \sin ^4(e+f x) \, dx &=\left (\sqrt [3]{\cos (e+f x)} \sqrt [3]{\sec (e+f x)}\right ) \int \frac{\sin ^4(e+f x)}{\cos ^{\frac{16}{3}}(e+f x)} \, dx\\ &=\frac{3 \, _2F_1\left (-\frac{13}{6},-\frac{3}{2};-\frac{7}{6};\cos ^2(e+f x)\right ) \sec ^{\frac{13}{3}}(e+f x) \sin (e+f x)}{13 f \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.10049, size = 89, normalized size = 1.68 \[ \frac{3 \sqrt [3]{\sec (e+f x)} \left (-18 \sin (e+f x) \sqrt [6]{\cos ^2(e+f x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )+27 \sin (e+f x)+\tan (e+f x) \sec (e+f x) \left (7 \sec ^2(e+f x)-16\right )\right )}{91 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ( \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 (\sec \left (f x + e\right )^{\frac{4}{3}} \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 \sec \left (f x + e\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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