Optimal. Leaf size=79 \[ \frac{2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]
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Rubi [A] time = 0.121835, antiderivative size = 79, 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 = {2787, 2785, 133} \[ \frac{2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]
Antiderivative was successfully verified.
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Rule 2787
Rule 2785
Rule 133
Rubi steps
\begin{align*} \int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx &=\frac{(a+a \cos (c+d x))^{2/3} \int \cos ^{\frac{4}{3}}(c+d x) (1+\cos (c+d x))^{2/3} \, dx}{(1+\cos (c+d x))^{2/3}}\\ &=\frac{\left ((a+a \cos (c+d x))^{2/3} \sin (c+d x)\right ) \operatorname{Subst}\left (\int \frac{(1-x)^{4/3} \sqrt [6]{2-x}}{\sqrt{x}} \, dx,x,1-\cos (c+d x)\right )}{d \sqrt{1-\cos (c+d x)} (1+\cos (c+d x))^{7/6}}\\ &=\frac{2 \sqrt [6]{2} F_1\left (\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right ) (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}}\\ \end{align*}
Mathematica [F] time = 3.28333, size = 0, normalized size = 0. \[ \int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.168, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}} \left ( a+\cos \left ( dx+c \right ) a \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{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}} \cos \left (d x + c\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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(-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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