3.71 \(\int \frac{x}{\sqrt{1-x^3} \left (4-x^3\right )} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{3\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{1-x^3}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{2} x+1}{\sqrt{1-x^3}}\right )}{3\ 2^{2/3}}+\frac{\tanh ^{-1}\left (\sqrt{1-x^3}\right )}{9\ 2^{2/3}} \]

[Out]

-ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]]/(3*2^(2/3)*Sqrt[3]) + ArcTan[Sq
rt[1 - x^3]/Sqrt[3]]/(3*2^(2/3)*Sqrt[3]) - ArcTanh[(1 + 2^(1/3)*x)/Sqrt[1 - x^3]
]/(3*2^(2/3)) + ArcTanh[Sqrt[1 - x^3]]/(9*2^(2/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.0675644, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{3\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{1-x^3}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{2} x+1}{\sqrt{1-x^3}}\right )}{3\ 2^{2/3}}+\frac{\tanh ^{-1}\left (\sqrt{1-x^3}\right )}{9\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(Sqrt[1 - x^3]*(4 - x^3)),x]

[Out]

-ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]]/(3*2^(2/3)*Sqrt[3]) + ArcTan[Sq
rt[1 - x^3]/Sqrt[3]]/(3*2^(2/3)*Sqrt[3]) - ArcTanh[(1 + 2^(1/3)*x)/Sqrt[1 - x^3]
]/(3*2^(2/3)) + ArcTanh[Sqrt[1 - x^3]]/(9*2^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.05896, size = 155, normalized size = 1.22 \[ \frac{\sqrt [3]{2} \log{\left (\sqrt [3]{2} x - \sqrt{- x^{3} + 1} + 1 \right )}}{12} - \frac{\sqrt [3]{2} \log{\left (\sqrt [3]{2} x + \sqrt{- x^{3} + 1} + 1 \right )}}{12} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \left (- \sqrt{- x^{3} + 1} + 1\right )}{3 x} \right )}}{18} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \left (\sqrt{- x^{3} + 1} + 1\right )}{3 x} \right )}}{18} + \frac{\sqrt [3]{2} \operatorname{atanh}{\left (\sqrt{- x^{3} + 1} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**3+4)/(-x**3+1)**(1/2),x)

[Out]

2**(1/3)*log(2**(1/3)*x - sqrt(-x**3 + 1) + 1)/12 - 2**(1/3)*log(2**(1/3)*x + sq
rt(-x**3 + 1) + 1)/12 - 2**(1/3)*sqrt(3)*atan(sqrt(3)/3 - 2**(2/3)*sqrt(3)*(-sqr
t(-x**3 + 1) + 1)/(3*x))/18 + 2**(1/3)*sqrt(3)*atan(sqrt(3)/3 - 2**(2/3)*sqrt(3)
*(sqrt(-x**3 + 1) + 1)/(3*x))/18 + 2**(1/3)*atanh(sqrt(-x**3 + 1))/18

_______________________________________________________________________________________

Mathematica [C]  time = 0.160891, size = 120, normalized size = 0.94 \[ -\frac{10 x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,\frac{x^3}{4}\right )}{\sqrt{1-x^3} \left (x^3-4\right ) \left (3 x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};x^3,\frac{x^3}{4}\right )+2 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};x^3,\frac{x^3}{4}\right )\right )+20 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,\frac{x^3}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/(Sqrt[1 - x^3]*(4 - x^3)),x]

[Out]

(-10*x^2*AppellF1[2/3, 1/2, 1, 5/3, x^3, x^3/4])/(Sqrt[1 - x^3]*(-4 + x^3)*(20*A
ppellF1[2/3, 1/2, 1, 5/3, x^3, x^3/4] + 3*x^3*(AppellF1[5/3, 1/2, 2, 8/3, x^3, x
^3/4] + 2*AppellF1[5/3, 3/2, 1, 8/3, x^3, x^3/4])))

_______________________________________________________________________________________

Maple [C]  time = 0.289, size = 164, normalized size = 1.3 \[{\frac{i}{36}}\sqrt{2}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}-4 \right ) }{{{\it \_alpha}}^{2} \left ( -2\,{{\it \_alpha}}^{2}+{\it \_alpha}+1+i\sqrt{3} \left ( 1-{\it \_alpha} \right ) \right ) \sqrt{{\frac{i}{2}} \left ( 2\,x+1-i\sqrt{3} \right ) }\sqrt{{\frac{-1+x}{i\sqrt{3}-3}}}\sqrt{-{\frac{i}{2}} \left ( 2\,x+1+i\sqrt{3} \right ) }{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{{\it \_alpha}}{2}}-{\frac{i}{3}}{{\it \_alpha}}^{2}\sqrt{3}-{\frac{1}{2}}+{\frac{i}{6}}{\it \_alpha}\,\sqrt{3}+{\frac{i}{6}}\sqrt{3},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^3+4)/(-x^3+1)^(1/2),x)

[Out]

1/36*I*2^(1/2)*sum(_alpha^2*(1/2*I*(2*x+1-I*3^(1/2)))^(1/2)*((-1+x)/(I*3^(1/2)-3
))^(1/2)*(-1/2*I*(2*x+1+I*3^(1/2)))^(1/2)/(-x^3+1)^(1/2)*(-2*_alpha^2+_alpha+1+I
*3^(1/2)*(1-_alpha))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1
/2),1/2*_alpha-1/3*I*_alpha^2*3^(1/2)-1/2+1/6*I*_alpha*3^(1/2)+1/6*I*3^(1/2),(I*
3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^3-4))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x}{{\left (x^{3} - 4\right )} \sqrt{-x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^3 - 4)*sqrt(-x^3 + 1)),x, algorithm="maxima")

[Out]

-integrate(x/((x^3 - 4)*sqrt(-x^3 + 1)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.358083, size = 1451, normalized size = 11.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^3 - 4)*sqrt(-x^3 + 1)),x, algorithm="fricas")

[Out]

1/15552*432^(5/6)*(sqrt(3)*log(2592*(1296*x^7 - 1296*x^4 + 6*2^(2/3)*(x^9 - 228*
x^6 + 264*x^3 - 64) + (72*x^7 - 1872*x^4 + 432^(5/6)*sqrt(3)*(7*x^5 - 4*x^2) - 1
44*432^(1/6)*sqrt(3)*(x^6 - x^3) + 1152*x)*sqrt(-x^3 + 1) - 216*2^(1/3)*(x^8 - 5
*x^5 + 4*x^2))/(x^9 - 12*x^6 + 48*x^3 - 64)) - sqrt(3)*log(2592*(1296*x^7 - 1296
*x^4 + 6*2^(2/3)*(x^9 - 228*x^6 + 264*x^3 - 64) - (72*x^7 - 1872*x^4 + 432^(5/6)
*sqrt(3)*(7*x^5 - 4*x^2) - 144*432^(1/6)*sqrt(3)*(x^6 - x^3) + 1152*x)*sqrt(-x^3
+ 1) - 216*2^(1/3)*(x^8 - 5*x^5 + 4*x^2))/(x^9 - 12*x^6 + 48*x^3 - 64)) + 8*arc
tan(-432*(18*x^5 + 2^(2/3)*(x^7 + 16*x^4 - 8*x) + 2^(1/3)*(5*x^6 + 20*x^3 - 16))
*sqrt(-x^3 + 1)/(432^(5/6)*(x^9 + 66*x^6 - 72*x^3 + 32) - 72*sqrt(3)*2^(1/3)*(x^
9 - 12*x^6 + 48*x^3 - 64) + 864*sqrt(3)*(x^8 + 7*x^5 - 8*x^2) + 1728*432^(1/6)*(
x^7 + x^4 - 2*x))) + 4*arctan(-216*(6*x^8 + 42*x^5 - 48*x^2 - 4*432^(1/6)*sqrt(3
)*(x^7 + x^4 - 2*x) - (18*x^5 + 2^(2/3)*(x^7 + 16*x^4 - 8*x) - 2*2^(1/3)*(5*x^6
+ 20*x^3 - 16))*sqrt(-x^3 + 1))/(36*sqrt(2)*(x^9 - 12*x^6 + 48*x^3 - 64)*sqrt((1
296*x^7 - 1296*x^4 + 6*2^(2/3)*(x^9 - 228*x^6 + 264*x^3 - 64) + (72*x^7 - 1872*x
^4 + 432^(5/6)*sqrt(3)*(7*x^5 - 4*x^2) - 144*432^(1/6)*sqrt(3)*(x^6 - x^3) + 115
2*x)*sqrt(-x^3 + 1) - 216*2^(1/3)*(x^8 - 5*x^5 + 4*x^2))/(x^9 - 12*x^6 + 48*x^3
- 64)) - 432^(5/6)*(x^9 + 66*x^6 - 72*x^3 + 32) + 432*sqrt(3)*(x^8 + 7*x^5 - 8*x
^2) + 216*(18*sqrt(3)*x^5 - sqrt(3)*2^(2/3)*(x^7 + 16*x^4 - 8*x))*sqrt(-x^3 + 1)
+ 864*432^(1/6)*(x^7 + x^4 - 2*x))) + 4*arctan(216*(6*x^8 + 42*x^5 - 48*x^2 - 4
*432^(1/6)*sqrt(3)*(x^7 + x^4 - 2*x) + (18*x^5 + 2^(2/3)*(x^7 + 16*x^4 - 8*x) -
2*2^(1/3)*(5*x^6 + 20*x^3 - 16))*sqrt(-x^3 + 1))/(36*sqrt(2)*(x^9 - 12*x^6 + 48*
x^3 - 64)*sqrt((1296*x^7 - 1296*x^4 + 6*2^(2/3)*(x^9 - 228*x^6 + 264*x^3 - 64) -
(72*x^7 - 1872*x^4 + 432^(5/6)*sqrt(3)*(7*x^5 - 4*x^2) - 144*432^(1/6)*sqrt(3)*
(x^6 - x^3) + 1152*x)*sqrt(-x^3 + 1) - 216*2^(1/3)*(x^8 - 5*x^5 + 4*x^2))/(x^9 -
12*x^6 + 48*x^3 - 64)) - 432^(5/6)*(x^9 + 66*x^6 - 72*x^3 + 32) + 432*sqrt(3)*(
x^8 + 7*x^5 - 8*x^2) - 216*(18*sqrt(3)*x^5 - sqrt(3)*2^(2/3)*(x^7 + 16*x^4 - 8*x
))*sqrt(-x^3 + 1) + 864*432^(1/6)*(x^7 + x^4 - 2*x))))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{x^{3} \sqrt{- x^{3} + 1} - 4 \sqrt{- x^{3} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**3+4)/(-x**3+1)**(1/2),x)

[Out]

-Integral(x/(x**3*sqrt(-x**3 + 1) - 4*sqrt(-x**3 + 1)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x}{{\left (x^{3} - 4\right )} \sqrt{-x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^3 - 4)*sqrt(-x^3 + 1)),x, algorithm="giac")

[Out]

integrate(-x/((x^3 - 4)*sqrt(-x^3 + 1)), x)