Optimal. Leaf size=58 \[ -\frac{3 b^3 \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{4}{3},\frac{7}{3},\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{8/3}} \]
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Rubi [A] time = 0.0432759, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {16, 3772, 2643} \[ -\frac{3 b^3 \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{8/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sqrt [3]{b \sec (c+d x)} \, dx &=b^2 \int \frac{1}{(b \sec (c+d x))^{5/3}} \, dx\\ &=\left (b^2 \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 )^{5/3} \, dx\\ &=-\frac{3 \cos ^3(c+d x) \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{8 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.143094, size = 59, normalized size = 1.02 \[ \frac{3 \sin (2 (c+d x)) \sqrt [3]{b \sec (c+d x)} \text{Hypergeometric2F1}\left (-\frac{5}{6},\frac{1}{2},\frac{1}{6},\sec ^2(c+d x)\right )}{10 d \sqrt{-\tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.166, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 )^{2}\,{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 )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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