Optimal. Leaf size=23 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.056197, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 - x^2)/((1 + x^2)*Sqrt[1 + x^4]),x]
[Out]
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Rubi in Sympy [A] time = 7.68499, size = 22, normalized size = 0.96 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)/(x**2+1)/(x**4+1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0646638, size = 40, normalized size = 1.74 \[ \sqrt [4]{-1} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-2 \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^2)/((1 + x^2)*Sqrt[1 + x^4]),x]
[Out]
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Maple [C] time = 0.023, size = 112, normalized size = 4.9 \[ -{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-2\,{\frac{ \left ( -1 \right ) ^{3/4}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,i,\sqrt{-i}- \left ( -1 \right ) ^{3/4} \right ) }{\sqrt{{x}^{4}+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)/(x^2+1)/(x^4+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 1}{\sqrt{x^{4} + 1}{\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(sqrt(x^4 + 1)*(x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241135, size = 24, normalized size = 1.04 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(sqrt(x^4 + 1)*(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{2}}{x^{2} \sqrt{x^{4} + 1} + \sqrt{x^{4} + 1}}\, dx - \int \left (- \frac{1}{x^{2} \sqrt{x^{4} + 1} + \sqrt{x^{4} + 1}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+1)/(x**2+1)/(x**4+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2} - 1}{\sqrt{x^{4} + 1}{\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)/(sqrt(x^4 + 1)*(x^2 + 1)),x, algorithm="giac")
[Out]