Optimal. Leaf size=149 \[ \frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{d \sqrt [3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{2 b d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.163131, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {16, 3021, 2748, 2643} \[ \frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{d \sqrt [3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{2 b d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3021
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx &=b \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx\\ &=\frac{3 A \sin (c+d x)}{d \sqrt [3]{b \cos (c+d x)}}+\frac{3 \int \frac{\frac{b^2 B}{3}-\frac{1}{3} b^2 (2 A-C) \cos (c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx}{b^2}\\ &=\frac{3 A \sin (c+d x)}{d \sqrt [3]{b \cos (c+d x)}}+B \int \frac{1}{\sqrt [3]{b \cos (c+d x)}} \, dx-\frac{(2 A-C) \int (b \cos (c+d x))^{2/3} \, dx}{b}\\ &=\frac{3 A \sin (c+d x)}{d \sqrt [3]{b \cos (c+d x)}}-\frac{3 B (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{2 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 (2 A-C) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b^2 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.24293, size = 268, normalized size = 1.8 \[ \frac{(b \cos (c+d x))^{2/3} (A \sec (c+d x)+B+C \cos (c+d x)) \left (\frac{4 (2 A-C) \sec (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \, _2F_1\left (-\frac{1}{2},-\frac{1}{6};\frac{5}{6};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right )}{\sqrt{\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )}}+\csc (c) \left (3 \sqrt{\sec ^2(c)} ((4 A-C) \cos (d x)-C \cos (2 c+d x))-5 (2 A-C) \sec (c) \cos \left (c-\tan ^{-1}(\tan (c))-d x\right )+(C-2 A) \sec (c) \cos \left (c+\tan ^{-1}(\tan (c))+d x\right )\right )-\frac{2 B \sqrt{\sec ^2(c)} \sin \left (2 d x-2 \tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{\sqrt [3]{\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )}}\right )}{2 b d \sqrt{\sec ^2(c)} (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.345, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sec \left ( dx+c \right ){\frac{1}{\sqrt [3]{b\cos \left ( dx+c \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac{1}{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} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )}{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} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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