Optimal. Leaf size=144 \[ \frac{(28 A+7 B+13 C) \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{14 \sqrt [6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac{3 (7 B-3 C) \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a}}{28 d}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{4/3}}{7 a d} \]
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Rubi [A] time = 0.178464, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {3023, 2751, 2652, 2651} \[ \frac{(28 A+7 B+13 C) \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right )}{14 \sqrt [6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac{3 (7 B-3 C) \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a}}{28 d}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{4/3}}{7 a d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \sqrt [3]{a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{3 C (a+a \cos (c+d x))^{4/3} \sin (c+d x)}{7 a d}+\frac{3 \int \sqrt [3]{a+a \cos (c+d x)} \left (\frac{1}{3} a (7 A+4 C)+\frac{1}{3} a (7 B-3 C) \cos (c+d x)\right ) \, dx}{7 a}\\ &=\frac{3 (7 B-3 C) \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{28 d}+\frac{3 C (a+a \cos (c+d x))^{4/3} \sin (c+d x)}{7 a d}+\frac{1}{28} (28 A+7 B+13 C) \int \sqrt [3]{a+a \cos (c+d x)} \, dx\\ &=\frac{3 (7 B-3 C) \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{28 d}+\frac{3 C (a+a \cos (c+d x))^{4/3} \sin (c+d x)}{7 a d}+\frac{\left ((28 A+7 B+13 C) \sqrt [3]{a+a \cos (c+d x)}\right ) \int \sqrt [3]{1+\cos (c+d x)} \, dx}{28 \sqrt [3]{1+\cos (c+d x)}}\\ &=\frac{3 (7 B-3 C) \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{28 d}+\frac{3 C (a+a \cos (c+d x))^{4/3} \sin (c+d x)}{7 a d}+\frac{(28 A+7 B+13 C) \sqrt [3]{a+a \cos (c+d x)} \, _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)}{14 \sqrt [6]{2} d (1+\cos (c+d x))^{5/6}}\\ \end{align*}
Mathematica [F] time = 0.117657, size = 0, normalized size = 0. \[ \int \sqrt [3]{a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.338, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{a+\cos \left ( dx+c \right ) a} \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \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 (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{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 (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{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 (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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