Optimal. Leaf size=142 \[ -\frac{3 b (A-2 C) \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{6},\frac{11}{6},\cos ^2(c+d x)\right )}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}-\frac{3 B \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\cos ^2(c+d x)\right )}{2 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}} \]
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Rubi [A] time = 0.135153, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {4047, 3772, 2643, 4046} \[ -\frac{3 b (A-2 C) \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}-\frac{3 B \sin (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{2 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{2/3}} \, dx &=\frac{B \int \sqrt [3]{b \sec (c+d x)} \, dx}{b}+\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{2/3}} \, dx\\ &=\frac{3 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}+(A-2 C) \int \frac{1}{(b \sec (c+d x))^{2/3}} \, dx+\frac{\left (B \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}{b}\\ &=-\frac{3 B \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)}{2 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}+\left ((A-2 C) \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 B \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)}{2 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 (A-2 C) \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 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}\\ \end{align*}
Mathematica [C] time = 1.83104, size = 173, normalized size = 1.22 \[ \frac{3 e^{-i d x} (\sin (d x)-i \cos (d x)) \sqrt [3]{b \sec (c+d x)} \left ((A-2 C) e^{i (c+d x)} \sqrt [3]{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{5}{3},-e^{2 i (c+d x)}\right )+4 B \sqrt [3]{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{3},\frac{7}{6},-e^{2 i (c+d x)}\right )-2 A \cos (c+d x)+4 i C \sin (c+d x)+4 C \cos (c+d x)\right )}{4 b d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.131, size = 0, normalized size = 0. \begin{align*} \int{(A+B\sec \left ( dx+c \right ) +C \left ( \sec \left ( dx+c \right ) \right ) ^{2}) \left ( b\sec \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}\, 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} + B \sec \left (d x + c\right ) + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{2}{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} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{1}{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 + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}}{\left (b \sec{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, 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} + B \sec \left (d x + c\right ) + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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