Optimal. Leaf size=151 \[ \frac{3 (8 A+5 C) \sin (c+d x) (b \sec (c+d x))^{2/3} \text{Hypergeometric2F1}\left (-\frac{1}{3},\frac{1}{2},\frac{2}{3},\cos ^2(c+d x)\right )}{16 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{5/3} \text{Hypergeometric2F1}\left (-\frac{5}{6},\frac{1}{2},\frac{1}{6},\cos ^2(c+d x)\right )}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{5/3}}{8 b d} \]
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Rubi [A] time = 0.160207, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {16, 4047, 3772, 2643, 4046} \[ \frac{3 (8 A+5 C) \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right )}{16 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{5/3} \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right )}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{5/3}}{8 b d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \sec (c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{\int (b \sec (c+d x))^{5/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx}{b}\\ &=\frac{\int (b \sec (c+d x))^{5/3} \left (A+C \sec ^2(c+d x)\right ) \, dx}{b}+\frac{B \int (b \sec (c+d x))^{8/3} \, dx}{b^2}\\ &=\frac{3 C (b \sec (c+d x))^{5/3} \tan (c+d x)}{8 b d}+\frac{(8 A+5 C) \int (b \sec (c+d x))^{5/3} \, dx}{8 b}+\frac{\left (B \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac{1}{\left (\frac{\cos (c+d x)}{b}\right )^{8/3}} \, dx}{b^2}\\ &=\frac{3 B \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/3} \sin (c+d x)}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C (b \sec (c+d x))^{5/3} \tan (c+d x)}{8 b d}+\frac{\left ((8 A+5 C) \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac{1}{\left (\frac{\cos (c+d x)}{b}\right )^{5/3}} \, dx}{8 b}\\ &=\frac{3 (8 A+5 C) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{16 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/3} \sin (c+d x)}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C (b \sec (c+d x))^{5/3} \tan (c+d x)}{8 b d}\\ \end{align*}
Mathematica [C] time = 5.63532, size = 265, normalized size = 1.75 \[ \frac{3 i \left (\frac{b e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{2/3} \left ((8 A+5 C) e^{i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{8/3} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{11}{6},-e^{2 i (c+d x)}\right )-16 B \left (1+e^{2 i (c+d x)}\right )^{8/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{4}{3},-e^{2 i (c+d x)}\right )-40 A e^{i (c+d x)}-80 A e^{3 i (c+d x)}-40 A e^{5 i (c+d x)}-16 B e^{4 i (c+d x)}+16 B-5 C e^{i (c+d x)}-70 C e^{3 i (c+d x)}-25 C e^{5 i (c+d x)}\right )}{40 \sqrt [3]{2} d \left (1+e^{2 i (c+d x)}\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.149, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( dx+c \right ) \left ( b\sec \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\sec \left ( dx+c \right ) +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} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{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 (C \sec \left (d x + c\right )^{3} + B \sec \left (d x + c\right )^{2} + A \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}}, 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 (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{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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