Optimal. Leaf size=135 \[ -\frac {9 C \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{28 d}+\frac {3 C (a+a \cos (c+d x))^{4/3} \sin (c+d x)}{7 a d}+\frac {(28 A+13 C) \sqrt [3]{a+a \cos (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{14 \sqrt [6]{2} d (1+\cos (c+d x))^{5/6}} \]
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Rubi [A]
time = 0.11, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3103, 2830,
2731, 2730} \begin {gather*} \frac {(28 A+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 C \sin (c+d x) (a \cos (c+d x)+a)^{4/3}}{7 a d}-\frac {9 C \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a}}{28 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rule 2731
Rule 2830
Rule 3103
Rubi steps
\begin {align*} \int \sqrt [3]{a+a \cos (c+d x)} \left (A+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)-a C \cos (c+d x)\right ) \, dx}{7 a}\\ &=-\frac {9 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+13 C) \int \sqrt [3]{a+a \cos (c+d x)} \, dx\\ &=-\frac {9 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+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 {9 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+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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.93, size = 240, normalized size = 1.78 \begin {gather*} \frac {3 \sqrt [3]{a (1+\cos (c+d x))} \left (-4 (28 A+13 C) \cot \left (\frac {c}{2}\right )+4 C \cos (d x) \sin (c)+\frac {(28 A+13 C) \csc \left (\frac {c}{4}\right ) \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 ) \sec \left (\frac {c}{4}\right ) \sqrt [3]{1+e^{i d x} \cos (c)+i e^{i d x} \sin (c)}}{\left (1+e^{i d x}\right ) \cos \left (\frac {c}{2}\right )+i \left (-1+e^{i d x}\right ) \sin \left (\frac {c}{2}\right )}+8 C \cos (2 d x) \sin (2 c)+4 C \cos (c) \sin (d x)+8 C \cos (2 c) \sin (2 d x)\right )}{112 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \left (a +a \cos \left (d x +c \right )\right )^{\frac {1}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt [3]{a \left (\cos {\left (c + d x \right )} + 1\right )} \left (A + C \cos ^{2}{\left (c + d x \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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