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 = 0.0606313, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 4357
Rule 266
Rule 43
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*}
Mathematica [C] time = 0.303941, size = 140, normalized size = 5.6 \[ -\frac{3}{40} \cos ^{\frac{5}{3}}(2 x)-\frac{3 e^{-6 i x} \sqrt [3]{1+e^{4 i x}} \left (2 e^{4 i x} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{4 i x}\right )+e^{8 i x} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{4 i x}\right )+\left (1+e^{4 i x}\right )^{2/3} \left (1+e^{8 i x}\right )\right )}{256\ 2^{2/3} \sqrt [3]{e^{-2 i x}+e^{2 i x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.17, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( x \right ) \right ) ^{4} \left ( \cos \left ( 2\,x \right ) \right ) ^{{\frac{2}{3}}}\tan \left ( x \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (2 \, x\right )^{\frac{2}{3}} \cos \left (x\right )^{4} \tan \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42886, size = 84, normalized size = 3.36 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18457, size = 34, normalized size = 1.36 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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