Optimal. Leaf size=165 \[ \frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (-3 m-2),\frac{1}{6} (4-3 m),\cos ^2(c+d x)\right )}{d (3 m+2) \sqrt{\sin ^2(c+d x)}}-\frac{3 A \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (1-3 m),\frac{1}{6} (7-3 m),\cos ^2(c+d x)\right )}{d (1-3 m) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.117112, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {20, 3787, 3772, 2643} \[ \frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-3 m-2);\frac{1}{6} (4-3 m);\cos ^2(c+d x)\right )}{d (3 m+2) \sqrt{\sin ^2(c+d x)}}-\frac{3 A \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (1-3 m);\frac{1}{6} (7-3 m);\cos ^2(c+d x)\right )}{d (1-3 m) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx &=\frac{(b \sec (c+d x))^{2/3} \int \sec ^{\frac{2}{3}+m}(c+d x) (A+B \sec (c+d x)) \, dx}{\sec ^{\frac{2}{3}}(c+d x)}\\ &=\frac{\left (A (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac{2}{3}+m}(c+d x) \, dx}{\sec ^{\frac{2}{3}}(c+d x)}+\frac{\left (B (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac{5}{3}+m}(c+d x) \, dx}{\sec ^{\frac{2}{3}}(c+d x)}\\ &=\left (A \cos ^{\frac{2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac{2}{3}-m}(c+d x) \, dx+\left (B \cos ^{\frac{2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac{5}{3}-m}(c+d x) \, dx\\ &=-\frac{3 A \, _2F_1\left (\frac{1}{2},\frac{1}{6} (1-3 m);\frac{1}{6} (7-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (1-3 m) \sqrt{\sin ^2(c+d x)}}+\frac{3 B \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-2-3 m);\frac{1}{6} (4-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.215456, size = 140, normalized size = 0.85 \[ \frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \left (A (3 m+5) \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (3 m+2),\frac{1}{6} (3 m+8),\sec ^2(c+d x)\right )+B (3 m+2) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (3 m+5),\frac{1}{6} (3 m+11),\sec ^2(c+d x)\right )\right )}{d (3 m+2) (3 m+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.143, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{m} \left ( b\sec \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\sec \left ( dx+c \right ) \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 (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{m}\,{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 (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{m}, 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 (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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