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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]