Optimal. Leaf size=114 \[ -\frac{3 A \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{5}{3},\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}-\frac{3 B \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{7}{6},\cos ^2(c+d x)\right )}{b d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0931583, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {16, 3787, 3772, 2643} \[ -\frac{3 A \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}-\frac{3 B \sin (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{b d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx &=\frac{\int \frac{A+B \sec (c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx}{b}\\ &=\frac{A \int \frac{1}{\sqrt [3]{b \sec (c+d x)}} \, dx}{b}+\frac{B \int (b \sec (c+d x))^{2/3} \, dx}{b^2}\\ &=\frac{\left (A \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{\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 )^{2/3}} \, dx}{b^2}\\ &=-\frac{3 B \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)}{b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 A \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)}}\\ \end{align*}
Mathematica [A] time = 0.076464, size = 91, normalized size = 0.8 \[ -\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \left (2 A \cos (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{2},\frac{5}{6},\sec ^2(c+d x)\right )-B \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\sec ^2(c+d x)\right )\right )}{2 b d \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.004, size = 0, normalized size = 0. \begin{align*} \int{\sec \left ( dx+c \right ) \left ( A+B\sec \left ( dx+c \right ) \right ) \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{{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\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 (\frac{{\left (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 )}, 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{\left (A + B \sec{\left (c + d x \right )}\right ) \sec{\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{{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\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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