Optimal. Leaf size=82 \[ -\frac{3 \sin (c+d x) \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (5-3 m),\frac{1}{6} (11-3 m),\cos ^2(c+d x)\right )}{d (5-3 m) \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]
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Rubi [A] time = 0.0406255, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {20, 3772, 2643} \[ -\frac{3 \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (5-3 m);\frac{1}{6} (11-3 m);\cos ^2(c+d x)\right )}{d (5-3 m) \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{\sec ^m(c+d x)}{(b \sec (c+d x))^{2/3}} \, dx &=\frac{\sec ^{\frac{2}{3}}(c+d x) \int \sec ^{-\frac{2}{3}+m}(c+d x) \, dx}{(b \sec (c+d x))^{2/3}}\\ &=\frac{\left (\cos ^{\frac{1}{3}+m}(c+d x) \sec ^{1+m}(c+d x)\right ) \int \cos ^{\frac{2}{3}-m}(c+d x) \, dx}{(b \sec (c+d x))^{2/3}}\\ &=-\frac{3 \, _2F_1\left (\frac{1}{2},\frac{1}{6} (5-3 m);\frac{1}{6} (11-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (5-3 m) (b \sec (c+d x))^{2/3} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.130089, size = 83, normalized size = 1.01 \[ \frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (m-\frac{2}{3}\right ),\frac{1}{2} \left (m+\frac{4}{3}\right ),\sec ^2(c+d x)\right )}{d \left (m-\frac{2}{3}\right ) (b \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.091, 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}}}}\, 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{\sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac{2}{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 )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{m}}{b \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{\sec ^{m}{\left (c + d x \right )}}{\left (b \sec{\left (c + d x \right )}\right )^{\frac{2}{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{\sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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