3.63 \(\int \frac{\sqrt{1-x^4}}{1+x^4} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0261983, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^4]/(1 + x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.40816, size = 36, normalized size = 0.73 \[ \frac{\operatorname{atan}{\left (\frac{x \left (x^{2} + 1\right )}{\sqrt{- x^{4} + 1}} \right )}}{2} + \frac{\operatorname{atanh}{\left (\frac{x \left (- x^{2} + 1\right )}{\sqrt{- x^{4} + 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x*(x**2 + 1)/sqrt(-x**4 + 1))/2 + atanh(x*(-x**2 + 1)/sqrt(-x**4 + 1))/2

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Mathematica [C]  time = 0.144058, size = 110, normalized size = 2.24 \[ -\frac{5 x \sqrt{1-x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};x^4,-x^4\right )}{\left (x^4+1\right ) \left (2 x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};x^4,-x^4\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};x^4,-x^4\right )\right )-5 F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};x^4,-x^4\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*x*Sqrt[1 - x^4]*AppellF1[1/4, -1/2, 1, 5/4, x^4, -x^4])/((1 + x^4)*(-5*Appel
lF1[1/4, -1/2, 1, 5/4, x^4, -x^4] + 2*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, x^4, -x
^4] + AppellF1[5/4, 1/2, 1, 9/4, x^4, -x^4])))

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Maple [B]  time = 0.022, size = 100, normalized size = 2. \[ -{\frac{1}{4}\arctan \left ({\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) }+{\frac{1}{4}\arctan \left ( -{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) }-{\frac{1}{8}\ln \left ({1 \left ({\frac{-{x}^{4}+1}{2\,{x}^{2}}}-{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) \left ({\frac{-{x}^{4}+1}{2\,{x}^{2}}}+{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)^(1/2)/(x^4+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.257111, size = 105, normalized size = 2.14 \[ \frac{1}{2} \, \arctan \left (-\frac{x^{3} + \sqrt{-x^{4} + 1} x^{2} + x}{x^{3} - x - \sqrt{-x^{4} + 1}}\right ) + \frac{1}{4} \, \log \left (-\frac{x^{4} - 2 \, x^{2} - 2 \, \sqrt{-x^{4} + 1} x - 1}{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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