Optimal. Leaf size=53 \[ -\frac{3 \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0244257, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {16, 2643} \[ -\frac{3 \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rubi steps
\begin{align*} \int \sqrt [3]{b \cos (c+d x)} \sec (c+d x) \, dx &=b \int \frac{1}{(b \cos (c+d x))^{2/3}} \, dx\\ &=-\frac{3 \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0484639, size = 54, normalized size = 1.02 \[ -\frac{3 b \sqrt{\sin ^2(c+d x)} \cot (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{d (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b\cos \left ( dx+c \right ) }\sec \left ( dx+c \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b \cos{\left (c + d x \right )}} \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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