Optimal. Leaf size=25 \[ -\frac {3}{64} \cos ^{\frac {8}{3}}(2 x)-\frac {3}{40} \cos ^{\frac {5}{3}}(2 x) \]
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Rubi [A] time = 0.06, 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 = {4357, 266, 43} \[ -\frac {3}{64} \cos ^{\frac {8}{3}}(2 x)-\frac {3}{40} \cos ^{\frac {5}{3}}(2 x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4357
Rubi steps
\begin {align*} \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx &=-\operatorname {Subst}\left (\int x^3 \left (-1+2 x^2\right )^{2/3} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int x (-1+2 x)^{2/3} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {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] time = 0.32, size = 140, normalized size = 5.60 \[ -\frac {3}{40} \cos ^{\frac {5}{3}}(2 x)-\frac {3 e^{-6 i x} \sqrt [3]{1+e^{4 i x}} \left (2 e^{4 i x} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-e^{4 i x}\right )+e^{8 i x} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-e^{4 i x}\right )+\left (1+e^{4 i x}\right )^{2/3} \left (1+e^{8 i x}\right )\right )}{256\ 2^{2/3} \sqrt [3]{e^{-2 i x}+e^{2 i x}}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^3(x) \cos ^{\frac {2}{3}}(2 x) \sin (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.71, size = 26, normalized size = 1.04 \[ -\frac {3}{320} \, {\left (20 \, \cos \relax (x)^{4} - 4 \, \cos \relax (x)^{2} - 3\right )} {\left (2 \, \cos \relax (x)^{2} - 1\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 25, normalized size = 1.00 \[ -\frac {3}{64} \, {\left (2 \, \cos \relax (x)^{2} - 1\right )}^{\frac {8}{3}} - \frac {3}{40} \, {\left (2 \, \cos \relax (x)^{2} - 1\right )}^{\frac {5}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[\int \left (\cos ^{4}\relax (x )\right ) \left (\cos ^{\frac {2}{3}}\left (2 x \right )\right ) \tan \relax (x )\, dx\]
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 \[ \int \cos \left (2 \, x\right )^{\frac {2}{3}} \cos \relax (x)^{4} \tan \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int {\cos \left (2\,x\right )}^{2/3}\,{\cos \relax (x)}^4\,\mathrm {tan}\relax (x) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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