Optimal. Leaf size=68 \[ \frac{\sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\cos (c+d x)+1} \sqrt [3]{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.0321445, antiderivative size = 68, 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} \[ \frac{\sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\cos (c+d x)+1} \sqrt [3]{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \frac{1}{(a+a \cos (c+d x))^{4/3}} \, dx &=\frac{\sqrt [3]{1+\cos (c+d x)} \int \frac{1}{(1+\cos (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \cos (c+d x)}}\\ &=\frac{\, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{2^{5/6} a d \sqrt [6]{1+\cos (c+d x)} \sqrt [3]{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0639312, size = 69, normalized size = 1.01 \[ \frac{6 \sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right )} \cot \left (\frac{1}{2} (c+d x)\right ) \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{5 d (a (\cos (c+d x)+1))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.129, 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 \frac{1}{{\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 (\frac{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cos{\left (c + d x \right )} + a\right )^{\frac{4}{3}}}\, dx \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{1}{{\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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