Optimal. Leaf size=144 \[ \frac {3 (A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x))^{2/3}}+\frac {3 C \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 a d}-\frac {(4 A-8 B+7 C) \sqrt [3]{a+a \cos (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{2 \sqrt [6]{2} a d (1+\cos (c+d x))^{5/6}} \]
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Rubi [A]
time = 0.14, antiderivative size = 144, normalized size of antiderivative = 1.00, 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 = {3102, 2829,
2731, 2730} \begin {gather*} -\frac {(4 A-8 B+7 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 )}{2 \sqrt [6]{2} a d (\cos (c+d x)+1)^{5/6}}+\frac {3 (A-B+C) \sin (c+d x)}{d (a \cos (c+d x)+a)^{2/3}}+\frac {3 C \sin (c+d x) \sqrt [3]{a \cos (c+d x)+a}}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rule 2731
Rule 2829
Rule 3102
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx &=\frac {3 C \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 a d}+\frac {3 \int \frac {\frac {1}{3} a (4 A+C)+\frac {1}{3} a (4 B-3 C) \cos (c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx}{4 a}\\ &=\frac {3 (A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x))^{2/3}}+\frac {3 C \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 a d}-\frac {(4 A-8 B+7 C) \int \sqrt [3]{a+a \cos (c+d x)} \, dx}{4 a}\\ &=\frac {3 (A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x))^{2/3}}+\frac {3 C \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 a d}-\frac {\left ((4 A-8 B+7 C) \sqrt [3]{a+a \cos (c+d x)}\right ) \int \sqrt [3]{1+\cos (c+d x)} \, dx}{4 a \sqrt [3]{1+\cos (c+d x)}}\\ &=\frac {3 (A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x))^{2/3}}+\frac {3 C \sqrt [3]{a+a \cos (c+d x)} \sin (c+d x)}{4 a d}-\frac {(4 A-8 B+7 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)}{2 \sqrt [6]{2} a d (1+\cos (c+d x))^{5/6}}\\ \end {align*}
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Mathematica [F]
time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )}{\left (a +a \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]
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 \frac {A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}}{\left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}}}\, 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 \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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