Optimal. Leaf size=56 \[ \frac{3 b \sin (c+d x) \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right )}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}} \]
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Rubi [A] time = 0.0373004, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {16, 2643} \[ \frac{3 b \sin (c+d x) \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right )}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx &=b^2 \int \frac{1}{(b \cos (c+d x))^{8/3}} \, dx\\ &=\frac{3 b \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0583944, size = 58, normalized size = 1.04 \[ \frac{3 b^2 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right )}{5 d (b \cos (c+d x))^{8/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( dx+c \right ) \right ) ^{2} \left ( b\cos \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 )^{2}}{\left (b \cos \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 \cos \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{2}}{b \cos \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 ^{2}{\left (c + d x \right )}}{\left (b \cos{\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 )^{2}}{\left (b \cos \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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