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