Optimal. Leaf size=90 \[ \frac{3 b (7 A+4 C) \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{2},\frac{5}{6},\cos ^2(c+d x)\right )}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 d} \]
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Rubi [A] time = 0.0838983, 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} \[ \frac{3 b (7 A+4 C) \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right )}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 d} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int (b \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{3 C (b \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}+\frac{1}{7} (7 A+4 C) \int (b \sec (c+d x))^{4/3} \, dx\\ &=\frac{3 C (b \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}+\frac{1}{7} \left ((7 A+4 C) \sqrt [3]{\frac{\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)}\right ) \int \frac{1}{\left (\frac{\cos (c+d x)}{b}\right )^{4/3}} \, dx\\ &=\frac{3 b (7 A+4 C) \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C (b \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}\\ \end{align*}
Mathematica [C] time = 1.03609, size = 182, normalized size = 2.02 \[ \frac{3 i e^{i (c+d x)} \cos ^3(c+d x) (b \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \left ((7 A+4 C) \left (1+e^{2 i (c+d x)}\right )^{7/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{5}{3},-e^{2 i (c+d x)}\right )-14 A \left (1+e^{2 i (c+d x)}\right )^2-4 C \left (5 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+1\right )\right )}{7 d \left (1+e^{2 i (c+d x)}\right )^2 (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sec \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}} \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{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 ({\left (C b \sec \left (d x + c\right )^{3} + A b \sec \left (d x + c\right )\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(-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} + A\right )} \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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