3.7 \(\int \frac{1}{\sqrt{-1+x^2} \left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{3 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )}{4 \sqrt{2}}-\frac{x \sqrt{x^2-1}}{4 \left (x^2+1\right )} \]

[Out]

-(x*Sqrt[-1 + x^2])/(4*(1 + x^2)) + (3*ArcTanh[(Sqrt[2]*x)/Sqrt[-1 + x^2]])/(4*S
qrt[2])

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Rubi [A]  time = 0.0388008, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )}{4 \sqrt{2}}-\frac{x \sqrt{x^2-1}}{4 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(x*Sqrt[-1 + x^2])/(4*(1 + x^2)) + (3*ArcTanh[(Sqrt[2]*x)/Sqrt[-1 + x^2]])/(4*S
qrt[2])

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Rubi in Sympy [A]  time = 3.71454, size = 41, normalized size = 0.85 \[ - \frac{x \sqrt{x^{2} - 1}}{4 \left (x^{2} + 1\right )} + \frac{3 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{x^{2} - 1}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(x**2 - 1)/(4*(x**2 + 1)) + 3*sqrt(2)*atanh(sqrt(2)*x/sqrt(x**2 - 1))/8

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Mathematica [A]  time = 0.0706282, size = 90, normalized size = 1.88 \[ -\frac{\sqrt{x^2-1} x}{4 \left (x^2+1\right )}+\frac{3 \log \left (-3 x^2-2 \sqrt{2} \sqrt{x^2-1} x+1\right )}{16 \sqrt{2}}-\frac{3 \log \left (-3 x^2+2 \sqrt{2} \sqrt{x^2-1} x+1\right )}{16 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-(x*Sqrt[-1 + x^2])/(4*(1 + x^2)) + (3*Log[1 - 3*x^2 - 2*Sqrt[2]*x*Sqrt[-1 + x^2
]])/(16*Sqrt[2]) - (3*Log[1 - 3*x^2 + 2*Sqrt[2]*x*Sqrt[-1 + x^2]])/(16*Sqrt[2])

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Maple [A]  time = 0.026, size = 45, normalized size = 0.9 \[ -{\frac{x}{8}{\frac{1}{\sqrt{{x}^{2}-1}}} \left ({\frac{{x}^{2}}{{x}^{2}-1}}-{\frac{1}{2}} \right ) ^{-1}}+{\frac{3\,\sqrt{2}}{8}{\it Artanh} \left ({x\sqrt{2}{\frac{1}{\sqrt{{x}^{2}-1}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/(x^2-1)^(1/2)*x/(x^2/(x^2-1)-1/2)+3/8*arctanh(x*2^(1/2)/(x^2-1)^(1/2))*2^(1
/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222009, size = 216, normalized size = 4.5 \[ -\frac{6 \, \sqrt{2} \sqrt{x^{2} - 1} x + 3 \,{\left (2 \, x^{4} + x^{2} - 2 \,{\left (x^{3} + x\right )} \sqrt{x^{2} - 1} - 1\right )} \log \left (\frac{4 \, x^{2} + \sqrt{2}{\left (2 \, x^{4} + x^{2} + 3\right )} - 2 \, \sqrt{x^{2} - 1}{\left (\sqrt{2}{\left (x^{3} + x\right )} + 2 \, x\right )} + 4}{2 \, x^{4} + x^{2} - 2 \,{\left (x^{3} + x\right )} \sqrt{x^{2} - 1} - 1}\right ) - 2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}}{8 \,{\left (2 \, \sqrt{2}{\left (x^{3} + x\right )} \sqrt{x^{2} - 1} - \sqrt{2}{\left (2 \, x^{4} + x^{2} - 1\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(6*sqrt(2)*sqrt(x^2 - 1)*x + 3*(2*x^4 + x^2 - 2*(x^3 + x)*sqrt(x^2 - 1) - 1
)*log((4*x^2 + sqrt(2)*(2*x^4 + x^2 + 3) - 2*sqrt(x^2 - 1)*(sqrt(2)*(x^3 + x) +
2*x) + 4)/(2*x^4 + x^2 - 2*(x^3 + x)*sqrt(x^2 - 1) - 1)) - 2*sqrt(2)*(3*x^2 - 1)
)/(2*sqrt(2)*(x^3 + x)*sqrt(x^2 - 1) - sqrt(2)*(2*x^4 + x^2 - 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.202751, size = 136, normalized size = 2.83 \[ -\frac{3}{16} \, \sqrt{2}{\rm ln}\left (\frac{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} - 2 \, \sqrt{2} + 3}{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 2 \, \sqrt{2} + 3}\right ) - \frac{3 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1}{2 \,{\left ({\left (x - \sqrt{x^{2} - 1}\right )}^{4} + 6 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/16*sqrt(2)*ln(((x - sqrt(x^2 - 1))^2 - 2*sqrt(2) + 3)/((x - sqrt(x^2 - 1))^2
+ 2*sqrt(2) + 3)) - 1/2*(3*(x - sqrt(x^2 - 1))^2 + 1)/((x - sqrt(x^2 - 1))^4 + 6
*(x - sqrt(x^2 - 1))^2 + 1)