Optimal. Leaf size=146 \[ -\frac{3 (A+4 C) \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{7}{6},\cos ^2(c+d x)\right )}{4 b d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}}-\frac{3 B \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{5}{3},\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}+\frac{3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}} \]
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Rubi [A] time = 0.138117, antiderivative size = 146, 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, 4045} \[ -\frac{3 (A+4 C) \sin (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{4 b d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}}+\frac{3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac{3 B \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3772
Rule 2643
Rule 4045
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx &=\frac{B \int \frac{1}{\sqrt [3]{b \sec (c+d x)}} \, dx}{b}+\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx\\ &=\frac{3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}+\frac{(A+4 C) \int (b \sec (c+d x))^{2/3} \, dx}{4 b^2}+\frac{\left (B \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}{b}\\ &=-\frac{3 B \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)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}+\frac{\left ((A+4 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 )^{2/3}} \, dx}{4 b^2}\\ &=-\frac{3 (A+4 C) \cos (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \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)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}\\ \end{align*}
Mathematica [C] time = 1.93147, size = 298, normalized size = 2.04 \[ \frac{\left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac{30 \cos (c+d x) (4 B \cot (c)-A \sin (c+d x))}{d}+\frac{3 i 2^{2/3} e^{-i d x} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{2/3} \left (1+e^{2 i (c+d x)}\right )^{2/3} \left (e^{i d x} \left (8 B e^{i (c+d x)} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{11}{6},-e^{2 i (c+d x)}\right )-5 \left (-1+e^{2 i c}\right ) (A+4 C) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{4}{3},-e^{2 i (c+d x)}\right )\right )+40 B e^{i c} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{2}{3},\frac{5}{6},-e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sec ^{\frac{2}{3}}(c+d x)}\right )}{20 (b \sec (c+d x))^{4/3} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.143, 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{4}{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{4}{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{2}{3}}}{b^{2} \sec \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] 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{4}{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{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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