Optimal. Leaf size=53 \[ -\frac {3 \sin (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(a+b x)\right )}{4 b \sqrt {\sin ^2(a+b x)} \sec ^{\frac {4}{3}}(a+b x)} \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3772, 2643} \[ -\frac {3 \sin (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(a+b x)\right )}{4 b \sqrt {\sin ^2(a+b x)} \sec ^{\frac {4}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{\sec (a+b x)}} \, dx &=\cos ^{\frac {2}{3}}(a+b x) \sec ^{\frac {2}{3}}(a+b x) \int \sqrt [3]{\cos (a+b x)} \, dx\\ &=-\frac {3 \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(a+b x)\right ) \sin (a+b x)}{4 b \sec ^{\frac {4}{3}}(a+b x) \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 53, normalized size = 1.00 \[ -\frac {3 \sqrt {-\tan ^2(a+b x)} \csc (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sec ^2(a+b x)\right )}{b \sec ^{\frac {4}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sec \left (b x + a\right )^{\frac {1}{3}}}, x\right ) \]
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 \[ \int \frac {1}{\sec \left (b x + a\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec \left (b x +a \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{\sec \left (b x + a\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\sqrt [3]{\sec {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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