Optimal. Leaf size=53 \[ -\frac {3 \sin (a+b x) \cos ^{\frac {5}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(a+b x)\right )}{5 b \sqrt {\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2643} \[ -\frac {3 \sin (a+b x) \cos ^{\frac {5}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(a+b x)\right )}{5 b \sqrt {\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin {align*} \int \cos ^{\frac {2}{3}}(a+b x) \, dx &=-\frac {3 \cos ^{\frac {5}{3}}(a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{5 b \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.00 \[ -\frac {3 \sqrt {\sin ^2(a+b x)} \cos ^{\frac {5}{3}}(a+b x) \csc (a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x + a\right )^{\frac {2}{3}}, x\right ) \]
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 \[ \int \cos \left (b x + a\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \cos ^{\frac {2}{3}}\left (b x +a \right )\, dx \]
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 \[ \int \cos \left (b x + a\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 42, normalized size = 0.79 \[ -\frac {3\,{\cos \left (a+b\,x\right )}^{5/3}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{6};\ \frac {11}{6};\ {\cos \left (a+b\,x\right )}^2\right )}{5\,b\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \]
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 \[ \int \cos ^{\frac {2}{3}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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