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

Optimal. Leaf size=73 \[ \frac{\sqrt{x^2-1} \sqrt{x^4-1} \sinh ^{-1}(x)}{\left (1-x^2\right ) \sqrt{x^2+1}}-\frac{\sqrt{x^4-1} \sin ^{-1}(x)}{\sqrt{1-x^2} \sqrt{x^2+1}} \]

[Out]

-((Sqrt[-1 + x^4]*ArcSin[x])/(Sqrt[1 - x^2]*Sqrt[1 + x^2])) + (Sqrt[-1 + x^2]*Sq
rt[-1 + x^4]*ArcSinh[x])/((1 - x^2)*Sqrt[1 + x^2])

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Rubi [A]  time = 0.204226, antiderivative size = 72, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right )}{\sqrt{x^4-1}}-\frac{\sqrt{x^2-1} \sqrt{x^2+1} \sinh ^{-1}(x)}{\sqrt{x^4-1}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[-1 + x^2]*Sqrt[1 + x^2]*ArcSinh[x])/Sqrt[-1 + x^4]) + (Sqrt[-1 + x^2]*Sq
rt[1 + x^2]*ArcTanh[x/Sqrt[-1 + x^2]])/Sqrt[-1 + x^4]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0212421, size = 71, normalized size = 0.97 \[ \log \left (1-x^2\right )-\log \left (x^2+1\right )-\log \left (x^3+\sqrt{x^2-1} \sqrt{x^4-1}-x\right )+\log \left (x^3+\sqrt{x^2+1} \sqrt{x^4-1}+x\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 - x^2] - Log[1 + x^2] - Log[-x + x^3 + Sqrt[-1 + x^2]*Sqrt[-1 + x^4]] + Lo
g[x + x^3 + Sqrt[1 + x^2]*Sqrt[-1 + x^4]]

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Maple [A]  time = 0.003, size = 59, normalized size = 0.8 \[ -{{\it Arcsinh} \left ( x \right ) \sqrt{{x}^{4}-1}{\frac{1}{\sqrt{{x}^{2}-1}}}{\frac{1}{\sqrt{{x}^{2}+1}}}}+{1\sqrt{{x}^{4}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.881393, size = 26, normalized size = 0.36 \[ -\operatorname{arsinh}\left (x\right ) + \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsinh(x) + log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 0.286901, size = 185, normalized size = 2.53 \[ \frac{1}{2} \, \log \left (\frac{x^{3} + \sqrt{x^{4} - 1} \sqrt{x^{2} + 1} + x}{x^{3} + x}\right ) - \frac{1}{2} \, \log \left (-\frac{x^{3} - \sqrt{x^{4} - 1} \sqrt{x^{2} + 1} + x}{x^{3} + x}\right ) - \frac{1}{2} \, \log \left (\frac{x^{3} + \sqrt{x^{4} - 1} \sqrt{x^{2} - 1} - x}{x^{3} - x}\right ) + \frac{1}{2} \, \log \left (-\frac{x^{3} - \sqrt{x^{4} - 1} \sqrt{x^{2} - 1} - x}{x^{3} - x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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