3.724 \(\int \frac{2 x-x^3+x^2 \sqrt{2-x^2}}{-2+2 x^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{x^2}{4}+\frac{1}{4} \sqrt{2-x^2} x+\frac{1}{4} \log \left (1-x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{2-x^2}}\right ) \]

[Out]

-x^2/4 + (x*Sqrt[2 - x^2])/4 - ArcTanh[x/Sqrt[2 - x^2]]/2 + Log[1 - x^2]/4

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Rubi [A]  time = 0.258306, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{x^2}{4}+\frac{1}{4} \sqrt{2-x^2} x+\frac{1}{4} \log \left (1-x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{2-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2*x - x^3 + x^2*Sqrt[2 - x^2])/(-2 + 2*x^2),x]

[Out]

-x^2/4 + (x*Sqrt[2 - x^2])/4 - ArcTanh[x/Sqrt[2 - x^2]]/2 + Log[1 - x^2]/4

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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((2*x-x**3+x**2*(-x**2+2)**(1/2))/(2*x**2-2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.042192, size = 77, normalized size = 1.43 \[ \frac{1}{4} \left (-x^2+\sqrt{2-x^2} x+\log \left (1-x^2\right )-\log \left (\sqrt{2-x^2}-x+2\right )+\log \left (\sqrt{2-x^2}+x+2\right )+\log (1-x)-\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2*x - x^3 + x^2*Sqrt[2 - x^2])/(-2 + 2*x^2),x]

[Out]

(-x^2 + x*Sqrt[2 - x^2] + Log[1 - x] - Log[1 + x] + Log[1 - x^2] - Log[2 - x + S
qrt[2 - x^2]] + Log[2 + x + Sqrt[2 - x^2]])/4

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Maple [B]  time = 0.019, size = 111, normalized size = 2.1 \[{\frac{x}{4}\sqrt{-{x}^{2}+2}}-{\frac{1}{4}\sqrt{- \left ( 1+x \right ) ^{2}+3+2\,x}}+{\frac{1}{4}{\it Artanh} \left ({\frac{4+2\,x}{2}{\frac{1}{\sqrt{- \left ( 1+x \right ) ^{2}+3+2\,x}}}} \right ) }+{\frac{1}{4}\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+3}}-{\frac{1}{4}{\it Artanh} \left ({\frac{-2\,x+4}{2}{\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+3}}}} \right ) }-{\frac{{x}^{2}}{4}}+{\frac{\ln \left ( -1+x \right ) }{4}}+{\frac{\ln \left ( 1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x*(-x^2+2)^(1/2)-1/4*(-(1+x)^2+3+2*x)^(1/2)+1/4*arctanh(1/2*(4+2*x)/(-(1+x)^
2+3+2*x)^(1/2))+1/4*(-(-1+x)^2-2*x+3)^(1/2)-1/4*arctanh(1/2*(-2*x+4)/(-(-1+x)^2-
2*x+3)^(1/2))-1/4*x^2+1/4*ln(-1+x)+1/4*ln(1+x)

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Maxima [A]  time = 0.766618, size = 127, normalized size = 2.35 \[ -\frac{1}{4} \, x^{2} + \frac{1}{4} \, \sqrt{-x^{2} + 2} x + \frac{1}{4} \, \log \left (x^{2} - 1\right ) + \frac{1}{4} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2}}{{\left | 2 \, x + 2 \right |}} + \frac{2}{{\left | 2 \, x + 2 \right |}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2}}{{\left | 2 \, x - 2 \right |}} + \frac{2}{{\left | 2 \, x - 2 \right |}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(x^3 - sqrt(-x^2 + 2)*x^2 - 2*x)/(x^2 - 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.281271, size = 90, normalized size = 1.67 \[ -\frac{1}{4} \, x^{2} + \frac{1}{4} \, \sqrt{-x^{2} + 2} x + \frac{1}{4} \, \log \left (x^{2} - 1\right ) - \frac{1}{8} \, \log \left (-\frac{\sqrt{-x^{2} + 2} x + 1}{x^{2}}\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{-x^{2} + 2} x - 1}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(x^3 - sqrt(-x^2 + 2)*x^2 - 2*x)/(x^2 - 1),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int \left (- \frac{2 x}{x^{2} - 1}\right )\, dx + \int \frac{x^{3}}{x^{2} - 1}\, dx + \int \left (- \frac{x^{2} \sqrt{- x^{2} + 2}}{x^{2} - 1}\right )\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(Integral(-2*x/(x**2 - 1), x) + Integral(x**3/(x**2 - 1), x) + Integral(-x**2*s
qrt(-x**2 + 2)/(x**2 - 1), x))/2

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GIAC/XCAS [A]  time = 0.282264, size = 158, normalized size = 2.93 \[ -\frac{1}{4} \, x^{2} + \frac{1}{4} \, \sqrt{-x^{2} + 2} x + \frac{1}{4} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | \frac{x}{\sqrt{2} - \sqrt{-x^{2} + 2}} - \frac{\sqrt{2} - \sqrt{-x^{2} + 2}}{x} + 2 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | \frac{x}{\sqrt{2} - \sqrt{-x^{2} + 2}} - \frac{\sqrt{2} - \sqrt{-x^{2} + 2}}{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(x^3 - sqrt(-x^2 + 2)*x^2 - 2*x)/(x^2 - 1),x, algorithm="giac")

[Out]

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