Optimal. Leaf size=56 \[ -\frac {3 \sin (a+b x) \sqrt [3]{c \cos (a+b x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(a+b x)\right )}{b c \sqrt {\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 \sin (a+b x) \sqrt [3]{c \cos (a+b x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(a+b x)\right )}{b c \sqrt {\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin {align*} \int \frac {1}{(c \cos (a+b x))^{2/3}} \, dx &=-\frac {3 \sqrt [3]{c \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 c \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.95 \[ -\frac {3 \sqrt {\sin ^2(a+b x)} \cot (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(a+b x)\right )}{b (c \cos (a+b x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c \cos \left (b x + a\right )\right )^{\frac {1}{3}}}{c \cos \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \cos \left (b x +a \right )\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \cos {\left (a + b x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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