Optimal. Leaf size=104 \[ \frac{2 \sqrt [6]{2} (5 A+2 B) \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{5 d (\cos (c+d x)+1)^{7/6}}+\frac{3 B \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 d} \]
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Rubi [A] time = 0.0819791, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2751, 2652, 2651} \[ \frac{2 \sqrt [6]{2} (5 A+2 B) \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{5 d (\cos (c+d x)+1)^{7/6}}+\frac{3 B \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx &=\frac{3 B (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{5 d}+\frac{1}{5} (5 A+2 B) \int (a+a \cos (c+d x))^{2/3} \, dx\\ &=\frac{3 B (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{5 d}+\frac{\left ((5 A+2 B) (a+a \cos (c+d x))^{2/3}\right ) \int (1+\cos (c+d x))^{2/3} \, dx}{5 (1+\cos (c+d x))^{2/3}}\\ &=\frac{3 B (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{5 d}+\frac{2 \sqrt [6]{2} (5 A+2 B) (a+a \cos (c+d x))^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{5 d (1+\cos (c+d x))^{7/6}}\\ \end{align*}
Mathematica [A] time = 0.591884, size = 164, normalized size = 1.58 \[ \frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a (\cos (c+d x)+1))^{2/3} \left (3\ 2^{5/6} \sin (c+d x) (5 A+2 B \cos (c+d x)+4 B) \sqrt [6]{1-\cos \left (d x-2 \tan ^{-1}\left (\cot \left (\frac{c}{2}\right )\right )\right )}-2 (5 A+2 B) \sin \left (d x-2 \tan ^{-1}\left (\cot \left (\frac{c}{2}\right )\right )\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2\left (\frac{d x}{2}-\tan ^{-1}\left (\cot \left (\frac{c}{2}\right )\right )\right )\right )\right )}{20\ 2^{5/6} d \sqrt [6]{1-\cos \left (d x-2 \tan ^{-1}\left (\cot \left (\frac{c}{2}\right )\right )\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.246, size = 0, normalized size = 0. \begin{align*} \int \left ( a+\cos \left ( dx+c \right ) a \right ) ^{{\frac{2}{3}}} \left ( A+B\cos \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 \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\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 ({\left (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}, 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 \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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