Optimal. Leaf size=102 \[ \frac {(4 A+B) \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right )}{2 \sqrt [6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac {3 B \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a}}{4 d} \]
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Rubi [A] time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2751, 2652, 2651} \[ \frac {(4 A+B) \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right )}{2 \sqrt [6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac {3 B \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a}}{4 d} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rule 2751
Rubi steps
\begin {align*} \int \sqrt [3]{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx &=\frac {3 B \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {1}{4} (4 A+B) \int \sqrt [3]{a+a \cos (c+d x)} \, dx\\ &=\frac {3 B \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {\left ((4 A+B) \sqrt [3]{a+a \cos (c+d x)}\right ) \int \sqrt [3]{1+\cos (c+d x)} \, dx}{4 \sqrt [3]{1+\cos (c+d x)}}\\ &=\frac {3 B \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {(4 A+B) \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)}{2 \sqrt [6]{2} d (1+\cos (c+d x))^{5/6}}\\ \end {align*}
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Mathematica [C] time = 3.53, size = 213, normalized size = 2.09 \[ \frac {3 \sqrt [3]{a (\cos (c+d x)+1)} \left (\frac {2 (4 A+B) \csc \left (\frac {c}{4}\right ) \sec \left (\frac {c}{4}\right ) \sqrt [3]{i \sin (c) e^{i d x}+\cos (c) e^{i d x}+1} \left (2 \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-e^{i d x} (\cos (c)+i \sin (c))\right )+e^{i d x} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-e^{i d x} (\cos (c)+i \sin (c))\right )\right )}{i \sin \left (\frac {c}{2}\right ) \left (-1+e^{i d x}\right )+\cos \left (\frac {c}{2}\right ) \left (1+e^{i d x}\right )}-8 (4 A+B) \cot \left (\frac {c}{2}\right )+8 B \sin (c) \cos (d x)+8 B \cos (c) \sin (d x)\right )}{32 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (a +a \cos \left (d x +c \right )\right )^{\frac {1}{3}} \left (A +B \cos \left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{a \left (\cos {\left (c + d x \right )} + 1\right )} \left (A + B \cos {\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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