Optimal. Leaf size=58 \[ -\frac{3 \sin (a+b x) (c \cos (a+b x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(a+b x)\right )}{2 b c \sqrt{\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.0153915, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ -\frac{3 \sin (a+b x) (c \cos (a+b x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(a+b x)\right )}{2 b c \sqrt{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{c \cos (a+b x)}} \, dx &=-\frac{3 (c \cos (a+b x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(a+b x)\right ) \sin (a+b x)}{2 b c \sqrt{\sin ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0413037, size = 55, normalized size = 0.95 \[ -\frac{3 \sqrt{\sin ^2(a+b x)} \cot (a+b x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(a+b x)\right )}{2 b \sqrt [3]{c \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{c\cos \left ( bx+a \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{1}{\left (c \cos \left (b x + a\right )\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{\left (c \cos \left (b x + a\right )\right )^{\frac{2}{3}}}{c \cos \left (b x + a\right )}, 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}{\sqrt [3]{c \cos{\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}{\left (c \cos \left (b x + a\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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