3.491 \(\int \frac{x^2}{1+\left (-1+x^2\right )^2} \, dx\)

Optimal. Leaf size=188 \[ \frac{\log \left (x^2-\sqrt{2 \left (1+\sqrt{2}\right )} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2 \left (1+\sqrt{2}\right )} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 x}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.397477, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ \frac{\log \left (x^2-\sqrt{2 \left (1+\sqrt{2}\right )} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2 \left (1+\sqrt{2}\right )} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 x}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 + (-1 + x^2)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 36.3409, size = 185, normalized size = 0.98 \[ \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x \sqrt{1 + \sqrt{2}} + \sqrt{2} \right )}}{8 \sqrt{1 + \sqrt{2}}} - \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x \sqrt{1 + \sqrt{2}} + \sqrt{2} \right )}}{8 \sqrt{1 + \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{4 \sqrt{-1 + \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{4 \sqrt{-1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log(x**2 - sqrt(2)*x*sqrt(1 + sqrt(2)) + sqrt(2))/(8*sqrt(1 + sqrt(2)))
- sqrt(2)*log(x**2 + sqrt(2)*x*sqrt(1 + sqrt(2)) + sqrt(2))/(8*sqrt(1 + sqrt(2))
) + sqrt(2)*atan(sqrt(2)*(x - sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + sqrt(2)))/(4*sqrt
(-1 + sqrt(2))) + sqrt(2)*atan(sqrt(2)*(x + sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + sqr
t(2)))/(4*sqrt(-1 + sqrt(2)))

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Mathematica [C]  time = 0.0456302, size = 39, normalized size = 0.21 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt{-1-i}}\right )}{(-1-i)^{3/2}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{-1+i}}\right )}{(-1+i)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(1 + (-1 + x^2)^2),x]

[Out]

-(ArcTan[x/Sqrt[-1 - I]]/(-1 - I)^(3/2)) - ArcTan[x/Sqrt[-1 + I]]/(-1 + I)^(3/2)

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Maple [B]  time = 0.047, size = 308, normalized size = 1.6 \[{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}\ln \left ({x}^{2}+\sqrt{2}-x\sqrt{2+2\,\sqrt{2}} \right ) }{8}}+{\frac{\sqrt{2} \left ( 2+2\,\sqrt{2} \right ) }{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\ln \left ({x}^{2}+\sqrt{2}-x\sqrt{2+2\,\sqrt{2}} \right ) }{8}}-{\frac{2+2\,\sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}\ln \left ({x}^{2}+\sqrt{2}+x\sqrt{2+2\,\sqrt{2}} \right ) }{8}}+{\frac{\sqrt{2} \left ( 2+2\,\sqrt{2} \right ) }{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}\ln \left ({x}^{2}+\sqrt{2}+x\sqrt{2+2\,\sqrt{2}} \right ) }{8}}-{\frac{2+2\,\sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(1+(x^2-1)^2),x)

[Out]

1/8*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(x^2+2^(1/2)-x*(2+2*2^(1/2))^(1/2))+1/4*2^(1/2
)*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*x-(2+2*2^(1/2))^(1/2))/(-2+2*2^(1
/2))^(1/2))-1/8*(2+2*2^(1/2))^(1/2)*ln(x^2+2^(1/2)-x*(2+2*2^(1/2))^(1/2))-1/4*(2
+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*x-(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))
^(1/2))-1/8*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(x^2+2^(1/2)+x*(2+2*2^(1/2))^(1/2))+1/
4*2^(1/2)*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*x+(2+2*2^(1/2))^(1/2))/(-
2+2*2^(1/2))^(1/2))+1/8*(2+2*2^(1/2))^(1/2)*ln(x^2+2^(1/2)+x*(2+2*2^(1/2))^(1/2)
)-1/4*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*x+(2+2*2^(1/2))^(1/2))/(-2+2*
2^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((x^2 - 1)^2 + 1), x)

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Fricas [A]  time = 0.28938, size = 630, normalized size = 3.35 \[ \frac{\sqrt{2}{\left (2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (24 \, \sqrt{2} x^{2} + 2^{\frac{3}{4}}{\left (17 \, \sqrt{2} x - 24 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 34 \, x^{2} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )}\right ) - 2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (24 \, \sqrt{2} x^{2} - 2^{\frac{3}{4}}{\left (17 \, \sqrt{2} x - 24 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 34 \, x^{2} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )}\right ) - 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{24 \, \sqrt{2} x^{2} + 2^{\frac{3}{4}}{\left (17 \, \sqrt{2} x - 24 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 34 \, x^{2} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )}}{12 \, \sqrt{2} - 17}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + \sqrt{2}{\left (\sqrt{2} x - x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 2^{\frac{1}{4}}}\right ) - 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{24 \, \sqrt{2} x^{2} - 2^{\frac{3}{4}}{\left (17 \, \sqrt{2} x - 24 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 34 \, x^{2} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )}}{12 \, \sqrt{2} - 17}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + \sqrt{2}{\left (\sqrt{2} x - x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 2^{\frac{1}{4}}}\right )\right )}}{8 \,{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(2^(1/4)*(sqrt(2) - 1)*log(24*sqrt(2)*x^2 + 2^(3/4)*(17*sqrt(2)*x -
24*x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 34*x^2 + 2*sqrt(2)*(12*sqrt(2) - 17)
) - 2^(1/4)*(sqrt(2) - 1)*log(24*sqrt(2)*x^2 - 2^(3/4)*(17*sqrt(2)*x - 24*x)*sqr
t((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 34*x^2 + 2*sqrt(2)*(12*sqrt(2) - 17)) - 4*2^(
1/4)*arctan(2^(1/4)*(sqrt(2) - 1)/(sqrt(2)*sqrt(1/2)*(sqrt(2) - 1)*sqrt((24*sqrt
(2)*x^2 + 2^(3/4)*(17*sqrt(2)*x - 24*x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 34
*x^2 + 2*sqrt(2)*(12*sqrt(2) - 17))/(12*sqrt(2) - 17))*sqrt((sqrt(2) - 2)/(2*sqr
t(2) - 3)) + sqrt(2)*(sqrt(2)*x - x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 2^(1/
4))) - 4*2^(1/4)*arctan(2^(1/4)*(sqrt(2) - 1)/(sqrt(2)*sqrt(1/2)*(sqrt(2) - 1)*s
qrt((24*sqrt(2)*x^2 - 2^(3/4)*(17*sqrt(2)*x - 24*x)*sqrt((sqrt(2) - 2)/(2*sqrt(2
) - 3)) - 34*x^2 + 2*sqrt(2)*(12*sqrt(2) - 17))/(12*sqrt(2) - 17))*sqrt((sqrt(2)
- 2)/(2*sqrt(2) - 3)) + sqrt(2)*(sqrt(2)*x - x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) -
3)) + 2^(1/4))))/((sqrt(2) - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)))

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Sympy [A]  time = 1.76616, size = 24, normalized size = 0.13 \[ \operatorname{RootSum}{\left (128 t^{4} + 16 t^{2} + 1, \left ( t \mapsto t \log{\left (64 t^{3} + 4 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(128*_t**4 + 16*_t**2 + 1, Lambda(_t, _t*log(64*_t**3 + 4*_t + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (x^{2} - 1\right )}^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((x^2 - 1)^2 + 1), x)