Optimal. Leaf size=58 \[ -\frac{3 b^2 \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{6},\frac{11}{6},\cos ^2(c+d x)\right )}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}} \]
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Rubi [A] time = 0.0358247, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {16, 3772, 2643} \[ -\frac{3 b^2 \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \cos (c+d x) \sqrt [3]{b \sec (c+d x)} \, dx &=b \int \frac{1}{(b \sec (c+d x))^{2/3}} \, dx\\ &=\left (b \sqrt [3]{\frac{\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)}\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{2/3} \, dx\\ &=-\frac{3 \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{5 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0669936, size = 58, normalized size = 1. \[ -\frac{3 b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{3},\frac{1}{2},\frac{2}{3},\sec ^2(c+d x)\right )}{2 d (b \sec (c+d x))^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.11, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \sqrt [3]{b\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 \sec \left (d x + c\right )\right )^{\frac{1}{3}} \cos \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 \sec \left (d x + c\right )\right )^{\frac{1}{3}} \cos \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 \sec{\left (c + d x \right )}} \cos{\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 \sec \left (d x + c\right )\right )^{\frac{1}{3}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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