Optimal. Leaf size=51 \[ -\frac{3 \sin (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{7}{6},\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)} \sqrt [3]{\sec (a+b x)}} \]
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Rubi [A] time = 0.0276046, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ -\frac{3 \sin (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)} \sqrt [3]{\sec (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^{\frac{2}{3}}(a+b x) \, dx &=\cos ^{\frac{2}{3}}(a+b x) \sec ^{\frac{2}{3}}(a+b x) \int \frac{1}{\cos ^{\frac{2}{3}}(a+b x)} \, dx\\ &=-\frac{3 \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt [3]{\sec (a+b x)} \sqrt{\sin ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0356083, size = 55, normalized size = 1.08 \[ \frac{3 \sqrt{-\tan ^2(a+b x)} \csc (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\sec ^2(a+b x)\right )}{2 b \sqrt [3]{\sec (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.094, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}\, 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 \sec \left (b x + a\right )^{\frac{2}{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 (\sec \left (b x + a\right )^{\frac{2}{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 \sec ^{\frac{2}{3}}{\left (a + b 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 \sec \left (b x + a\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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