Optimal. Leaf size=92 \[ \frac{3 (7 A+4 C) \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{2},\frac{5}{6},\cos ^2(c+d x)\right )}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 b d} \]
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Rubi [A] time = 0.0801193, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {16, 4046, 3772, 2643} \[ \frac{3 (7 A+4 C) \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right )}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec (c+d x) \sqrt [3]{b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{\int (b \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \, dx}{b}\\ &=\frac{3 C (b \sec (c+d x))^{4/3} \tan (c+d x)}{7 b d}+\frac{(7 A+4 C) \int (b \sec (c+d x))^{4/3} \, dx}{7 b}\\ &=\frac{3 C (b \sec (c+d x))^{4/3} \tan (c+d x)}{7 b d}+\frac{\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}{7 b}\\ &=\frac{3 (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 b d}\\ \end{align*}
Mathematica [C] time = 1.50716, size = 185, normalized size = 2.01 \[ \frac{3 i e^{i (c+d x)} \cos ^3(c+d x) (b \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \left ((7 A+4 C) \left (1+e^{2 i (c+d x)}\right )^{7/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{5}{3},-e^{2 i (c+d x)}\right )-14 A \left (1+e^{2 i (c+d x)}\right )^2-4 C \left (5 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+1\right )\right )}{7 b d \left (1+e^{2 i (c+d x)}\right )^2 (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.134, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( dx+c \right ) \sqrt [3]{b\sec \left ( dx+c \right ) } \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{1}{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} + A \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b \sec{\left (c + d x \right )}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \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{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{1}{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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