Optimal. Leaf size=25 \[ -\frac {3}{40} \cos ^{\frac {5}{3}}(2 x)-\frac {3}{64} \cos ^{\frac {8}{3}}(2 x) \]
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Rubi [A]
time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4442, 272, 45}
\begin {gather*} -\frac {3}{64} \cos ^{\frac {8}{3}}(2 x)-\frac {3}{40} \cos ^{\frac {5}{3}}(2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4442
Rubi steps
\begin {align*} \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx &=-\text {Subst}\left (\int x^3 \left (-1+2 x^2\right )^{2/3} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int x (-1+2 x)^{2/3} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{2} (-1+2 x)^{2/3}+\frac {1}{2} (-1+2 x)^{5/3}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=-\frac {3}{40} \left (-1+2 \cos ^2(x)\right )^{5/3}-\frac {3}{64} \left (-1+2 \cos ^2(x)\right )^{8/3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 8.25, size = 140, normalized size = 5.60 \begin {gather*} -\frac {3}{40} \cos ^{\frac {5}{3}}(2 x)-\frac {3 e^{-6 i x} \sqrt [3]{1+e^{4 i x}} \left (\left (1+e^{4 i x}\right )^{2/3} \left (1+e^{8 i x}\right )+2 e^{4 i x} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-e^{4 i x}\right )+e^{8 i x} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-e^{4 i x}\right )\right )}{256\ 2^{2/3} \sqrt [3]{e^{-2 i x}+e^{2 i x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (\cos ^{4}\left (x \right )\right ) \left (\cos ^{\frac {2}{3}}\left (2 x \right )\right ) \tan \left (x \right )\, dx\]
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 [A]
time = 0.77, size = 26, normalized size = 1.04 \begin {gather*} -\frac {3}{320} \, {\left (20 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} - 3\right )} {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A]
time = 0.97, size = 25, normalized size = 1.00 \begin {gather*} -\frac {3}{64} \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {8}{3}} - \frac {3}{40} \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {5}{3}} \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 = -0.04 \begin {gather*} \int {\cos \left (2\,x\right )}^{2/3}\,{\cos \left (x\right )}^4\,\mathrm {tan}\left (x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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