3.806 \(\int x \sqrt{-2+x^4} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{4} x^2 \sqrt{x^4-2}-\frac{1}{2} \tanh ^{-1}\left (\frac{x^2}{\sqrt{x^4-2}}\right ) \]

[Out]

(x^2*Sqrt[-2 + x^4])/4 - ArcTanh[x^2/Sqrt[-2 + x^4]]/2

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Rubi [A]  time = 0.0290756, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{1}{4} x^2 \sqrt{x^4-2}-\frac{1}{2} \tanh ^{-1}\left (\frac{x^2}{\sqrt{x^4-2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[-2 + x^4],x]

[Out]

(x^2*Sqrt[-2 + x^4])/4 - ArcTanh[x^2/Sqrt[-2 + x^4]]/2

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Rubi in Sympy [A]  time = 2.52855, size = 27, normalized size = 0.77 \[ \frac{x^{2} \sqrt{x^{4} - 2}}{4} - \frac{\operatorname{atanh}{\left (\frac{x^{2}}{\sqrt{x^{4} - 2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(x**4 - 2)/4 - atanh(x**2/sqrt(x**4 - 2))/2

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Mathematica [A]  time = 0.0148069, size = 35, normalized size = 1. \[ \frac{1}{4} x^2 \sqrt{x^4-2}-\frac{1}{2} \log \left (\sqrt{x^4-2}+x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[-2 + x^4],x]

[Out]

(x^2*Sqrt[-2 + x^4])/4 - Log[x^2 + Sqrt[-2 + x^4]]/2

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Maple [A]  time = 0.012, size = 28, normalized size = 0.8 \[{\frac{{x}^{2}}{4}\sqrt{{x}^{4}-2}}-{\frac{1}{2}\ln \left ({x}^{2}+\sqrt{{x}^{4}-2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(x^4-2)^(1/2),x)

[Out]

1/4*x^2*(x^4-2)^(1/2)-1/2*ln(x^2+(x^4-2)^(1/2))

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Maxima [A]  time = 1.44101, size = 78, normalized size = 2.23 \[ -\frac{\sqrt{x^{4} - 2}}{2 \, x^{2}{\left (\frac{x^{4} - 2}{x^{4}} - 1\right )}} - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} - 2}}{x^{2}} + 1\right ) + \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} - 2}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 - 2)*x,x, algorithm="maxima")

[Out]

-1/2*sqrt(x^4 - 2)/(x^2*((x^4 - 2)/x^4 - 1)) - 1/4*log(sqrt(x^4 - 2)/x^2 + 1) +
1/4*log(sqrt(x^4 - 2)/x^2 - 1)

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Fricas [A]  time = 0.273931, size = 109, normalized size = 3.11 \[ -\frac{x^{8} - 2 \, x^{4} - 2 \,{\left (x^{4} - \sqrt{x^{4} - 2} x^{2} - 1\right )} \log \left (-x^{2} + \sqrt{x^{4} - 2}\right ) -{\left (x^{6} - x^{2}\right )} \sqrt{x^{4} - 2}}{4 \,{\left (x^{4} - \sqrt{x^{4} - 2} x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 - 2)*x,x, algorithm="fricas")

[Out]

-1/4*(x^8 - 2*x^4 - 2*(x^4 - sqrt(x^4 - 2)*x^2 - 1)*log(-x^2 + sqrt(x^4 - 2)) -
(x^6 - x^2)*sqrt(x^4 - 2))/(x^4 - sqrt(x^4 - 2)*x^2 - 1)

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Sympy [A]  time = 5.27351, size = 90, normalized size = 2.57 \[ \begin{cases} \frac{x^{6}}{4 \sqrt{x^{4} - 2}} - \frac{x^{2}}{2 \sqrt{x^{4} - 2}} - \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} x^{2}}{2} \right )}}{2} & \text{for}\: \frac{\left |{x^{4}}\right |}{2} > 1 \\- \frac{i x^{6}}{4 \sqrt{- x^{4} + 2}} + \frac{i x^{2}}{2 \sqrt{- x^{4} + 2}} + \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} x^{2}}{2} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(x**4-2)**(1/2),x)

[Out]

Piecewise((x**6/(4*sqrt(x**4 - 2)) - x**2/(2*sqrt(x**4 - 2)) - acosh(sqrt(2)*x**
2/2)/2, Abs(x**4)/2 > 1), (-I*x**6/(4*sqrt(-x**4 + 2)) + I*x**2/(2*sqrt(-x**4 +
2)) + I*asin(sqrt(2)*x**2/2)/2, True))

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GIAC/XCAS [A]  time = 0.219372, size = 39, normalized size = 1.11 \[ \frac{1}{4} \, \sqrt{x^{4} - 2} x^{2} + \frac{1}{2} \,{\rm ln}\left (x^{2} - \sqrt{x^{4} - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 - 2)*x,x, algorithm="giac")

[Out]

1/4*sqrt(x^4 - 2)*x^2 + 1/2*ln(x^2 - sqrt(x^4 - 2))