Optimal. Leaf size=105 \[ \frac{\sqrt{2} \sin (c+d x) \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}} \]
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Rubi [A] time = 0.0642733, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2665, 139, 138} \[ \frac{\sqrt{2} \sin (c+d x) \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2665
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cos (c+d x))^{2/3}} \, dx &=-\frac{\sin (c+d x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{1+x} (a+b x)^{2/3}} \, dx,x,\cos (c+d x)\right )}{d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)}}\\ &=-\frac{\left (\left (-\frac{a+b \cos (c+d x)}{-a-b}\right )^{2/3} \sin (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{1+x} \left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^{2/3}} \, dx,x,\cos (c+d x)\right )}{d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)} (a+b \cos (c+d x))^{2/3}}\\ &=\frac{\sqrt{2} F_1\left (\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right ) \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3} \sin (c+d x)}{d \sqrt{1+\cos (c+d x)} (a+b \cos (c+d x))^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.175637, size = 116, normalized size = 1.1 \[ -\frac{3 \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \sqrt [3]{a+b \cos (c+d x)} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right )}{b d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.205, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( dx+c \right ) \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 \frac{1}{{\left (b \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 (\frac{1}{{\left (b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \cos{\left (c + d x \right )}\right )^{\frac{2}{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 (b \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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