3.1380 \(\int x^2 \sqrt{-2+x^6} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{6} x^3 \sqrt{x^6-2}-\frac{1}{3} \tanh ^{-1}\left (\frac{x^3}{\sqrt{x^6-2}}\right ) \]

[Out]

(x^3*Sqrt[-2 + x^6])/6 - ArcTanh[x^3/Sqrt[-2 + x^6]]/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0319141, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{6} x^3 \sqrt{x^6-2}-\frac{1}{3} \tanh ^{-1}\left (\frac{x^3}{\sqrt{x^6-2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[-2 + x^6],x]

[Out]

(x^3*Sqrt[-2 + x^6])/6 - ArcTanh[x^3/Sqrt[-2 + x^6]]/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.88427, size = 27, normalized size = 0.77 \[ \frac{x^{3} \sqrt{x^{6} - 2}}{6} - \frac{\operatorname{atanh}{\left (\frac{x^{3}}{\sqrt{x^{6} - 2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(x**6-2)**(1/2),x)

[Out]

x**3*sqrt(x**6 - 2)/6 - atanh(x**3/sqrt(x**6 - 2))/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0141157, size = 35, normalized size = 1. \[ \frac{1}{6} x^3 \sqrt{x^6-2}-\frac{1}{3} \log \left (\sqrt{x^6-2}+x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[-2 + x^6],x]

[Out]

(x^3*Sqrt[-2 + x^6])/6 - Log[x^3 + Sqrt[-2 + x^6]]/3

_______________________________________________________________________________________

Maple [C]  time = 0.069, size = 47, normalized size = 1.3 \[{\frac{{x}^{3}}{6}\sqrt{{x}^{6}-2}}-{\frac{1}{3}\sqrt{-{\it signum} \left ( -1+{\frac{{x}^{6}}{2}} \right ) }\arcsin \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{\it signum} \left ( -1+{\frac{{x}^{6}}{2}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(x^6-2)^(1/2),x)

[Out]

1/6*x^3*(x^6-2)^(1/2)-1/3/signum(-1+1/2*x^6)^(1/2)*(-signum(-1+1/2*x^6))^(1/2)*a
rcsin(1/2*x^3*2^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.43738, size = 78, normalized size = 2.23 \[ -\frac{\sqrt{x^{6} - 2}}{3 \, x^{3}{\left (\frac{x^{6} - 2}{x^{6}} - 1\right )}} - \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} - 2}}{x^{3}} + 1\right ) + \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} - 2}}{x^{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^6 - 2)*x^2,x, algorithm="maxima")

[Out]

-1/3*sqrt(x^6 - 2)/(x^3*((x^6 - 2)/x^6 - 1)) - 1/6*log(sqrt(x^6 - 2)/x^3 + 1) +
1/6*log(sqrt(x^6 - 2)/x^3 - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.225602, size = 109, normalized size = 3.11 \[ -\frac{x^{12} - 2 \, x^{6} - 2 \,{\left (x^{6} - \sqrt{x^{6} - 2} x^{3} - 1\right )} \log \left (-x^{3} + \sqrt{x^{6} - 2}\right ) -{\left (x^{9} - x^{3}\right )} \sqrt{x^{6} - 2}}{6 \,{\left (x^{6} - \sqrt{x^{6} - 2} x^{3} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^6 - 2)*x^2,x, algorithm="fricas")

[Out]

-1/6*(x^12 - 2*x^6 - 2*(x^6 - sqrt(x^6 - 2)*x^3 - 1)*log(-x^3 + sqrt(x^6 - 2)) -
(x^9 - x^3)*sqrt(x^6 - 2))/(x^6 - sqrt(x^6 - 2)*x^3 - 1)

_______________________________________________________________________________________

Sympy [A]  time = 5.47264, size = 90, normalized size = 2.57 \[ \begin{cases} \frac{x^{9}}{6 \sqrt{x^{6} - 2}} - \frac{x^{3}}{3 \sqrt{x^{6} - 2}} - \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} & \text{for}\: \frac{\left |{x^{6}}\right |}{2} > 1 \\- \frac{i x^{9}}{6 \sqrt{- x^{6} + 2}} + \frac{i x^{3}}{3 \sqrt{- x^{6} + 2}} + \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(x**6-2)**(1/2),x)

[Out]

Piecewise((x**9/(6*sqrt(x**6 - 2)) - x**3/(3*sqrt(x**6 - 2)) - acosh(sqrt(2)*x**
3/2)/3, Abs(x**6)/2 > 1), (-I*x**9/(6*sqrt(-x**6 + 2)) + I*x**3/(3*sqrt(-x**6 +
2)) + I*asin(sqrt(2)*x**3/2)/3, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223445, size = 41, normalized size = 1.17 \[ \frac{1}{6} \, \sqrt{x^{6} - 2} x^{3} + \frac{1}{3} \,{\rm ln}\left ({\left | -x^{3} + \sqrt{x^{6} - 2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^6 - 2)*x^2,x, algorithm="giac")

[Out]

1/6*sqrt(x^6 - 2)*x^3 + 1/3*ln(abs(-x^3 + sqrt(x^6 - 2)))