Optimal. Leaf size=79 \[ \frac {2 \sqrt [6]{2} F_1\left (\frac {1}{2};-\frac {4}{3},-\frac {1}{6};\frac {3}{2};1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}} \]
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Rubi [A]
time = 0.09, antiderivative size = 79, 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 = {2866, 2864,
138} \begin {gather*} \frac {2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left (\frac {1}{2};-\frac {4}{3},-\frac {1}{6};\frac {3}{2};1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 2864
Rule 2866
Rubi steps
\begin {align*} \int \cos ^{\frac {4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx &=\frac {(a+a \cos (c+d x))^{2/3} \int \cos ^{\frac {4}{3}}(c+d x) (1+\cos (c+d x))^{2/3} \, dx}{(1+\cos (c+d x))^{2/3}}\\ &=\frac {\left ((a+a \cos (c+d x))^{2/3} \sin (c+d x)\right ) \text {Subst}\left (\int \frac {(1-x)^{4/3} \sqrt [6]{2-x}}{\sqrt {x}} \, dx,x,1-\cos (c+d x)\right )}{d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{7/6}}\\ &=\frac {2 \sqrt [6]{2} F_1\left (\frac {1}{2};-\frac {4}{3},-\frac {1}{6};\frac {3}{2};1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) (a+a \cos (c+d x))^{2/3} \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}}\\ \end {align*}
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Mathematica [F]
time = 3.33, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^{\frac {4}{3}}(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 = 0.08, size = 0, normalized size = 0.00 \[\int \left (\cos ^{\frac {4}{3}}\left (d x +c \right )\right ) \left (a +a \cos \left (d x +c \right )\right )^{\frac {2}{3}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\cos \left (c+d\,x\right )}^{4/3}\,{\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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