Optimal. Leaf size=90 \[ \frac{3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}-\frac{3 b (2 A-C) \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{5}{3},\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
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Rubi [A] time = 0.0739019, antiderivative size = 90, 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} \[ \frac{3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}-\frac{3 b (2 A-C) \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx &=\frac{3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}+\frac{1}{2} (2 A-C) \int \frac{1}{\sqrt [3]{b \sec (c+d x)}} \, dx\\ &=\frac{3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}+\frac{1}{2} \left ((2 A-C) \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \sqrt [3]{\frac{\cos (c+d x)}{b}} \, dx\\ &=-\frac{3 (2 A-C) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{8 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x)}{2 d \sqrt [3]{b \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.724211, size = 127, normalized size = 1.41 \[ -\frac{3 i \left ((2 A-C) e^{2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{11}{6},-e^{2 i (c+d x)}\right )-5 \left (A e^{2 i (c+d x)}+A-C e^{2 i (c+d x)}\right )\right )}{5 d \left (1+e^{2 i (c+d x)}\right ) \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{(A+C \left ( \sec \left ( dx+c \right ) \right ) ^{2}){\frac{1}{\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 \frac{C \sec \left (d x + c\right )^{2} + A}{\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 (\frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}}}{b \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 \frac{A + C \sec ^{2}{\left (c + d x \right )}}{\sqrt [3]{b \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 \frac{C \sec \left (d x + c\right )^{2} + A}{\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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