Optimal. Leaf size=93 \[ \frac{3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac{3 (2 A+5 C) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{5 b d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.101386, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {16, 3012, 2643} \[ \frac{3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac{3 (2 A+5 C) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{5 b d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3012
Rule 2643
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx &=b^2 \int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{8/3}} \, dx\\ &=\frac{3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}+\frac{1}{5} (2 A+5 C) \int \frac{1}{(b \cos (c+d x))^{2/3}} \, dx\\ &=\frac{3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac{3 (2 A+5 C) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.675028, size = 103, normalized size = 1.11 \[ \frac{6 A \tan (c+d x) \sqrt [6]{\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )}-(2 A+5 C) \sin \left (2 d x-2 \tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{10 d (b \cos (c+d x))^{2/3} \sqrt [6]{\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.352, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \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 (C \cos \left (d x + c\right )^{2} + A\right )} \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(-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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \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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