Optimal. Leaf size=53 \[ \frac{3 \sin (e+f x) \sec ^{\frac{2}{3}}(e+f x) \, _2F_1\left (-\frac{1}{2},-\frac{1}{3};\frac{2}{3};\cos ^2(e+f x)\right )}{2 f \sqrt{\sin ^2(e+f x)}} \]
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Rubi [A] time = 0.0579557, 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} \[ \frac{3 \sin (e+f x) \sec ^{\frac{2}{3}}(e+f x) \, _2F_1\left (-\frac{1}{2},-\frac{1}{3};\frac{2}{3};\cos ^2(e+f x)\right )}{2 f \sqrt{\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2632
Rule 2576
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{3}}(e+f x) \sin ^2(e+f x) \, dx &=\left (\cos ^{\frac{2}{3}}(e+f x) \sec ^{\frac{2}{3}}(e+f x)\right ) \int \frac{\sin ^2(e+f x)}{\cos ^{\frac{5}{3}}(e+f x)} \, dx\\ &=\frac{3 \, _2F_1\left (-\frac{1}{2},-\frac{1}{3};\frac{2}{3};\cos ^2(e+f x)\right ) \sec ^{\frac{2}{3}}(e+f x) \sin (e+f x)}{2 f \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0804503, size = 56, normalized size = 1.06 \[ -\frac{3 \sin (e+f x) \sec ^{\frac{2}{3}}(e+f x) \left (\sqrt [3]{\cos ^2(e+f x)} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )-1\right )}{2 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tan \left ( fx+e \right ) \right ) ^{2}{\frac{1}{\sqrt [3]{\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 )^{2}}{\sec \left (f x + e\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{\tan \left (f x + e\right )^{2}}{\sec \left (f x + e\right )^{\frac{1}{3}}}, 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 ^{2}{\left (e + f x \right )}}{\sqrt [3]{\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 )^{2}}{\sec \left (f x + e\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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