Optimal. Leaf size=51 \[ \frac {3 \sin (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right )}{b \sqrt {\sin ^2(a+b x)} \sqrt [3]{\cos (a+b x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 51, 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} \[ \frac {3 \sin (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right )}{b \sqrt {\sin ^2(a+b x)} \sqrt [3]{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {4}{3}}(a+b x)} \, dx &=\frac {3 \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt [3]{\cos (a+b x)} \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.00 \[ \frac {3 \sqrt {\sin ^2(a+b x)} \csc (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(a+b x)\right )}{b \sqrt [3]{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cos \left (b x + a\right )^{\frac {4}{3}}}, x\right ) \]
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 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \left (b x +a \right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 42, normalized size = 0.82 \[ \frac {3\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{6},\frac {1}{2};\ \frac {5}{6};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\cos \left (a+b\,x\right )}^{1/3}\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \]
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 \[ \int \frac {1}{\cos ^{\frac {4}{3}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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