Optimal. Leaf size=67 \[ \frac{2\ 2^{5/6} a \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{5/6}} \]
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Rubi [A] time = 0.0359337, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2652, 2651} \[ \frac{2\ 2^{5/6} a \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{5/6}} \]
Antiderivative was successfully verified.
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Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{4/3} \, dx &=\frac{\left (a \sqrt [3]{a+a \cos (c+d x)}\right ) \int (1+\cos (c+d x))^{4/3} \, dx}{\sqrt [3]{1+\cos (c+d x)}}\\ &=\frac{2\ 2^{5/6} a \sqrt [3]{a+a \cos (c+d x)} \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d (1+\cos (c+d x))^{5/6}}\\ \end{align*}
Mathematica [A] time = 0.0997721, size = 69, normalized size = 1.03 \[ -\frac{6 \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) (a (\cos (c+d x)+1))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{11 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.133, size = 0, normalized size = 0. \begin{align*} \int \left ( a+\cos \left ( dx+c \right ) a \right ) ^{{\frac{4}{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{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{4}{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 (a \cos \left (d x + c\right ) + a\right )}^{\frac{4}{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 (a \cos \left (d x + c\right ) + a\right )}^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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