Optimal. Leaf size=53 \[ -\frac {3 \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(a+b x)\right ) \sin (a+b x)}{2 b \sec ^{\frac {2}{3}}(a+b x) \sqrt {\sin ^2(a+b x)}} \]
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Rubi [A]
time = 0.02, 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 = {3857, 2722}
\begin {gather*} -\frac {3 \sin (a+b x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(a+b x)\right )}{2 b \sqrt {\sin ^2(a+b x)} \sec ^{\frac {2}{3}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3857
Rubi steps
\begin {align*} \int \sqrt [3]{\sec (a+b x)} \, dx &=\sqrt [3]{\cos (a+b x)} \sqrt [3]{\sec (a+b x)} \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx\\ &=-\frac {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 \sec ^{\frac {2}{3}}(a+b x) \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 53, normalized size = 1.00 \begin {gather*} \frac {3 \csc (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sec ^2(a+b x)\right ) \sqrt {-\tan ^2(a+b x)}}{b \sec ^{\frac {2}{3}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \sec ^{\frac {1}{3}}\left (b x +a \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt [3]{\sec {\left (a + b x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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