Optimal. Leaf size=88 \[ \frac{3 C \tan (c+d x) \sqrt [3]{b \sec (c+d x)}}{4 d}-\frac{3 b (4 A+C) \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]
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Rubi [A] time = 0.0641493, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {4046, 3772, 2643} \[ \frac{3 C \tan (c+d x) \sqrt [3]{b \sec (c+d x)}}{4 d}-\frac{3 b (4 A+C) \sin (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sqrt [3]{b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{3 C \sqrt [3]{b \sec (c+d x)} \tan (c+d x)}{4 d}+\frac{1}{4} (4 A+C) \int \sqrt [3]{b \sec (c+d x)} \, dx\\ &=\frac{3 C \sqrt [3]{b \sec (c+d x)} \tan (c+d x)}{4 d}+\frac{1}{4} \left ((4 A+C) \sqrt [3]{\frac{\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt [3]{\frac{\cos (c+d x)}{b}}} \, dx\\ &=-\frac{3 (4 A+C) \cos (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{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)}}+\frac{3 C \sqrt [3]{b \sec (c+d x)} \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [C] time = 1.17849, size = 162, normalized size = 1.84 \[ \frac{3 \sqrt [3]{b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (C \sin (c+d x) \sec ^{\frac{4}{3}}(c+d x)-i \sqrt [3]{2} (4 A+C) \sqrt [3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt [3]{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{3},\frac{7}{6},-e^{2 i (c+d x)}\right )\right )}{2 d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b\sec \left ( dx+c \right ) } \left ( A+C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \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 (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}}\,{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 (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}}, 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 )}} \left (A + C \sec ^{2}{\left (c + d x \right )}\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 (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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