3.324 \(\int \frac{1+x^2}{\left (1-x^2\right ) \sqrt{1+x^2+x^4}} \, dx\)

Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3} x}{\sqrt{x^4+x^2+1}}\right )}{\sqrt{3}} \]

[Out]

ArcTanh[(Sqrt[3]*x)/Sqrt[1 + x^2 + x^4]]/Sqrt[3]

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Rubi [A]  time = 0.0702955, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{3} x}{\sqrt{x^4+x^2+1}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^2)/((1 - x^2)*Sqrt[1 + x^2 + x^4]),x]

[Out]

ArcTanh[(Sqrt[3]*x)/Sqrt[1 + x^2 + x^4]]/Sqrt[3]

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Rubi in Sympy [A]  time = 8.17769, size = 26, normalized size = 1. \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} x}{\sqrt{x^{4} + x^{2} + 1}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*atanh(sqrt(3)*x/sqrt(x**4 + x**2 + 1))/3

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Mathematica [A]  time = 0.241202, size = 0, normalized size = 0. \[ \int \frac{1+x^2}{\left (1-x^2\right ) \sqrt{1+x^2+x^4}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(1 + x^2)/((1 - x^2)*Sqrt[1 + x^2 + x^4]),x]

[Out]

Integrate[(1 + x^2)/((1 - x^2)*Sqrt[1 + x^2 + x^4]), x]

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Maple [C]  time = 0.244, size = 184, normalized size = 7.1 \[ -2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{3}},1/2\,\sqrt{-2+2\,i\sqrt{3}} \right ) }{\sqrt{-2+2\,i\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}}+2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{3} \right ){x}^{2}}}{\sqrt{-1/2+i/2\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}{\it EllipticPi} \left ( \sqrt{-1/2+i/2\sqrt{3}}x, \left ( -1/2+i/2\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-1/2-i/2\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1
/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2
+2*I*3^(1/2))^(1/2))+2/(-1/2+1/2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2)^(
1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*EllipticPi((-1/2+1/2*I
*3^(1/2))^(1/2)*x,1/(-1/2+1/2*I*3^(1/2)),(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I*
3^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} + 1}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 + 1)/(sqrt(x^4 + x^2 + 1)*(x^2 - 1)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.231322, size = 63, normalized size = 2.42 \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{\sqrt{3}{\left (x^{4} + 4 \, x^{2} + 1\right )} + 6 \, \sqrt{x^{4} + x^{2} + 1} x}{x^{4} - 2 \, x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 + 1)/(sqrt(x^4 + x^2 + 1)*(x^2 - 1)),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log((sqrt(3)*(x^4 + 4*x^2 + 1) + 6*sqrt(x^4 + x^2 + 1)*x)/(x^4 - 2*x
^2 + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{2}}{x^{2} \sqrt{x^{4} + x^{2} + 1} - \sqrt{x^{4} + x^{2} + 1}}\, dx - \int \frac{1}{x^{2} \sqrt{x^{4} + x^{2} + 1} - \sqrt{x^{4} + x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x**2/(x**2*sqrt(x**4 + x**2 + 1) - sqrt(x**4 + x**2 + 1)), x) - Integr
al(1/(x**2*sqrt(x**4 + x**2 + 1) - sqrt(x**4 + x**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2} + 1}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 + 1)/(sqrt(x^4 + x^2 + 1)*(x^2 - 1)),x, algorithm="giac")

[Out]

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