3.454 \(\int \cos ^3(x) \cos ^{\frac{2}{3}}(2 x) \sin (x) \, dx\)
Optimal. Leaf size=25 \[ -\frac{3}{64} \cos ^{\frac{8}{3}}(2 x)-\frac{3}{40} \cos ^{\frac{5}{3}}(2 x) \]
[Out]
(-3*Cos[2*x]^(5/3))/40 - (3*Cos[2*x]^(8/3))/64
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Rubi [A] time = 0.0976713, 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
\[ -\frac{3}{64} \cos ^{\frac{8}{3}}(2 x)-\frac{3}{40} \cos ^{\frac{5}{3}}(2 x) \]
Antiderivative was successfully verified.
[In] Int[Cos[x]^3*Cos[2*x]^(2/3)*Sin[x],x]
[Out]
(-3*Cos[2*x]^(5/3))/40 - (3*Cos[2*x]^(8/3))/64
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(x)**4*cos(2*x)**(2/3)*tan(x),x)
[Out]
Timed out
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Mathematica [C] time = 0.519398, 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.
[In] Integrate[Cos[x]^3*Cos[2*x]^(2/3)*Sin[x],x]
[Out]
(-3*Cos[2*x]^(5/3))/40 - (3*(1 + E^((4*I)*x))^(1/3)*((1 + E^((4*I)*x))^(2/3)*(1
+ E^((8*I)*x)) + 2*E^((4*I)*x)*Hypergeometric2F1[-1/3, 1/3, 2/3, -E^((4*I)*x)] +
E^((8*I)*x)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((4*I)*x)]))/(256*2^(2/3)*E^((6
*I)*x)*(E^((-2*I)*x) + E^((2*I)*x))^(1/3))
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Maple [F] time = 0.254, size = 0, normalized size = 0. \[ \int \left ( \cos \left ( x \right ) \right ) ^{4} \left ( \cos \left ( 2\,x \right ) \right ) ^{{\frac{2}{3}}}\tan \left ( x \right ) \, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(x)^4*cos(2*x)^(2/3)*tan(x),x)
[Out]
int(cos(x)^4*cos(2*x)^(2/3)*tan(x),x)
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Maxima [A] time = 1.96459, size = 1392, normalized size = 55.68 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(2*x)^(2/3)*cos(x)^4*tan(x),x, algorithm="maxima")
[Out]
-3/2560*2^(1/3)*(((5*cos(16/3*x)*cos(3/2*arctan2(sin(16/3*x), cos(16/3*x))) + 16
*cos(16/3*x)*cos(9/8*arctan2(sin(16/3*x), cos(16/3*x))) + 10*cos(16/3*x)*cos(3/4
*arctan2(sin(16/3*x), cos(16/3*x))) + 16*cos(16/3*x)*cos(3/8*arctan2(sin(16/3*x)
, cos(16/3*x))) + 5*sin(16/3*x)*sin(3/2*arctan2(sin(16/3*x), cos(16/3*x))) + 16*
sin(16/3*x)*sin(9/8*arctan2(sin(16/3*x), cos(16/3*x))) + 10*sin(16/3*x)*sin(3/4*
arctan2(sin(16/3*x), cos(16/3*x))) + 16*sin(16/3*x)*sin(3/8*arctan2(sin(16/3*x),
cos(16/3*x))) + 5*cos(16/3*x))*cos(2/3*arctan2(sin(1/4*arctan2(sin(16/3*x), cos
(16/3*x))), cos(1/4*arctan2(sin(16/3*x), cos(16/3*x))) + 1)) + (5*cos(3/2*arctan
2(sin(16/3*x), cos(16/3*x)))*sin(16/3*x) + 16*cos(9/8*arctan2(sin(16/3*x), cos(1
6/3*x)))*sin(16/3*x) + 10*cos(3/4*arctan2(sin(16/3*x), cos(16/3*x)))*sin(16/3*x)
+ 16*cos(3/8*arctan2(sin(16/3*x), cos(16/3*x)))*sin(16/3*x) - 5*cos(16/3*x)*sin
(3/2*arctan2(sin(16/3*x), cos(16/3*x))) - 16*cos(16/3*x)*sin(9/8*arctan2(sin(16/
3*x), cos(16/3*x))) - 10*cos(16/3*x)*sin(3/4*arctan2(sin(16/3*x), cos(16/3*x)))
- 16*cos(16/3*x)*sin(3/8*arctan2(sin(16/3*x), cos(16/3*x))) + 5*sin(16/3*x))*sin
(2/3*arctan2(sin(1/4*arctan2(sin(16/3*x), cos(16/3*x))), cos(1/4*arctan2(sin(16/
3*x), cos(16/3*x))) + 1)))*cos(2/3*arctan2(sin(1/2*arctan2(sin(16/3*x), cos(16/3
*x))) - sin(1/4*arctan2(sin(16/3*x), cos(16/3*x))), cos(1/2*arctan2(sin(16/3*x),
cos(16/3*x))) - cos(1/4*arctan2(sin(16/3*x), cos(16/3*x))) + 1)) + ((5*cos(3/2*
arctan2(sin(16/3*x), cos(16/3*x)))*sin(16/3*x) + 16*cos(9/8*arctan2(sin(16/3*x),
cos(16/3*x)))*sin(16/3*x) + 10*cos(3/4*arctan2(sin(16/3*x), cos(16/3*x)))*sin(1
6/3*x) + 16*cos(3/8*arctan2(sin(16/3*x), cos(16/3*x)))*sin(16/3*x) - 5*cos(16/3*
x)*sin(3/2*arctan2(sin(16/3*x), cos(16/3*x))) - 16*cos(16/3*x)*sin(9/8*arctan2(s
in(16/3*x), cos(16/3*x))) - 10*cos(16/3*x)*sin(3/4*arctan2(sin(16/3*x), cos(16/3
*x))) - 16*cos(16/3*x)*sin(3/8*arctan2(sin(16/3*x), cos(16/3*x))) + 5*sin(16/3*x
))*cos(2/3*arctan2(sin(1/4*arctan2(sin(16/3*x), cos(16/3*x))), cos(1/4*arctan2(s
in(16/3*x), cos(16/3*x))) + 1)) - (5*cos(16/3*x)*cos(3/2*arctan2(sin(16/3*x), co
s(16/3*x))) + 16*cos(16/3*x)*cos(9/8*arctan2(sin(16/3*x), cos(16/3*x))) + 10*cos
(16/3*x)*cos(3/4*arctan2(sin(16/3*x), cos(16/3*x))) + 16*cos(16/3*x)*cos(3/8*arc
tan2(sin(16/3*x), cos(16/3*x))) + 5*sin(16/3*x)*sin(3/2*arctan2(sin(16/3*x), cos
(16/3*x))) + 16*sin(16/3*x)*sin(9/8*arctan2(sin(16/3*x), cos(16/3*x))) + 10*sin(
16/3*x)*sin(3/4*arctan2(sin(16/3*x), cos(16/3*x))) + 16*sin(16/3*x)*sin(3/8*arct
an2(sin(16/3*x), cos(16/3*x))) + 5*cos(16/3*x))*sin(2/3*arctan2(sin(1/4*arctan2(
sin(16/3*x), cos(16/3*x))), cos(1/4*arctan2(sin(16/3*x), cos(16/3*x))) + 1)))*si
n(2/3*arctan2(sin(1/2*arctan2(sin(16/3*x), cos(16/3*x))) - sin(1/4*arctan2(sin(1
6/3*x), cos(16/3*x))), cos(1/2*arctan2(sin(16/3*x), cos(16/3*x))) - cos(1/4*arct
an2(sin(16/3*x), cos(16/3*x))) + 1)))*(-2*(cos(1/4*arctan2(sin(16/3*x), cos(16/3
*x))) - 1)*cos(1/2*arctan2(sin(16/3*x), cos(16/3*x))) + cos(1/2*arctan2(sin(16/3
*x), cos(16/3*x)))^2 + cos(1/4*arctan2(sin(16/3*x), cos(16/3*x)))^2 + sin(1/2*ar
ctan2(sin(16/3*x), cos(16/3*x)))^2 - 2*sin(1/2*arctan2(sin(16/3*x), cos(16/3*x))
)*sin(1/4*arctan2(sin(16/3*x), cos(16/3*x))) + sin(1/4*arctan2(sin(16/3*x), cos(
16/3*x)))^2 - 2*cos(1/4*arctan2(sin(16/3*x), cos(16/3*x))) + 1)^(1/3)*(cos(1/4*a
rctan2(sin(16/3*x), cos(16/3*x)))^2 + sin(1/4*arctan2(sin(16/3*x), cos(16/3*x)))
^2 + 2*cos(1/4*arctan2(sin(16/3*x), cos(16/3*x))) + 1)^(1/3)
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Fricas [A] time = 0.223395, size = 35, normalized size = 1.4 \[ -\frac{3}{320} \,{\left (20 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} - 3\right )}{\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac{2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(2*x)^(2/3)*cos(x)^4*tan(x),x, algorithm="fricas")
[Out]
-3/320*(20*cos(x)^4 - 4*cos(x)^2 - 3)*(2*cos(x)^2 - 1)^(2/3)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)**4*cos(2*x)**(2/3)*tan(x),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.220223, size = 34, normalized size = 1.36 \[ -\frac{3}{64} \,{\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac{8}{3}} - \frac{3}{40} \,{\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac{5}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(2*x)^(2/3)*cos(x)^4*tan(x),x, algorithm="giac")
[Out]
-3/64*(2*cos(x)^2 - 1)^(8/3) - 3/40*(2*cos(x)^2 - 1)^(5/3)