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

Optimal. Leaf size=33 \[ \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0396453, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00501797, size = 56, normalized size = 1.7 \[ \frac{1}{2} \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \left (\log \left (\frac{x}{\sqrt{x^2-1}}+1\right )-\log \left (1-\frac{x}{\sqrt{x^2-1}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(-1 + x^2)^(-1)]*Sqrt[-1 + x^2]*(-Log[1 - x/Sqrt[-1 + x^2]] + Log[1 + x/Sq
rt[-1 + x^2]]))/2

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Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \[ \sqrt{ \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((x^2+1)/(x^4-1))^(1/2),x)

[Out]

(1/(x^2-1))^(1/2)*(x^2-1)^(1/2)*ln(x+(x^2-1)^(1/2))

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Maxima [A]  time = 0.71637, size = 19, normalized size = 0.58 \[ \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 0.266031, size = 31, normalized size = 0.94 \[ -\log \left (-\sqrt{x^{2} - 1}{\left (\frac{x}{\sqrt{x^{2} - 1}} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-sqrt(x^2 - 1)*(x/sqrt(x^2 - 1) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264541, size = 28, normalized size = 0.85 \[ -{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ){\rm sign}\left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(-x + sqrt(x^2 - 1)))*sign(x^2 - 1)