Optimal. Leaf size=51 \[ -\frac{3 \sin (a+b x) \sqrt [3]{\cos (a+b x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.0118621, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2643} \[ -\frac{3 \sin (a+b x) \sqrt [3]{\cos (a+b x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b \sqrt{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{2}{3}}(a+b x)} \, dx &=-\frac{3 \sqrt [3]{\cos (a+b x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt{\sin ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0247452, size = 51, normalized size = 1. \[ -\frac{3 \sqrt{\sin ^2(a+b x)} \sqrt [3]{\cos (a+b x)} \csc (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.139, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \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 \frac{1}{\cos \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 (\frac{1}{\cos \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 \frac{1}{\cos ^{\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 \frac{1}{\cos \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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