Optimal. Leaf size=34 \[ \frac{x \log \left (\sqrt{x^2+1}+x\right )}{\sqrt{1-x^2}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]
[Out]
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Rubi [A] time = 0.0675983, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \log \left (\sqrt{x^2+1}+x\right )}{\sqrt{1-x^2}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
[In] Int[Log[x + Sqrt[1 + x^2]]/(1 - x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 5.87164, size = 27, normalized size = 0.79 \[ \frac{x \log{\left (x + \sqrt{x^{2} + 1} \right )}}{\sqrt{- x^{2} + 1}} - \frac{\operatorname{asin}{\left (x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(ln(x+(x**2+1)**(1/2))/(-x**2+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.133861, size = 64, normalized size = 1.88 \[ \frac{1}{2} \sqrt{1-x^2} \left (-\frac{2 x \log \left (\sqrt{x^2+1}+x\right )}{x^2-1}-\frac{\sqrt{x^2+1} \sin ^{-1}\left (x^2\right )}{\sqrt{1-x^4}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Log[x + Sqrt[1 + x^2]]/(1 - x^2)^(3/2),x]
[Out]
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Maple [F] time = 0.029, size = 0, normalized size = 0. \[ \int{1\ln \left ( x+\sqrt{{x}^{2}+1} \right ) \left ( -{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(ln(x+(x^2+1)^(1/2))/(-x^2+1)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\log \left (x + \sqrt{x^{2} + 1}\right )}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(x + sqrt(x^2 + 1))/(-x^2 + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229682, size = 84, normalized size = 2.47 \[ -\frac{\sqrt{-x^{2} + 1} x \log \left (x + \sqrt{x^{2} + 1}\right ) -{\left (x^{2} - 1\right )} \arctan \left (\frac{\sqrt{x^{2} + 1} \sqrt{-x^{2} + 1} - 1}{x^{2}}\right )}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(x + sqrt(x^2 + 1))/(-x^2 + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(ln(x+(x**2+1)**(1/2))/(-x**2+1)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213412, size = 49, normalized size = 1.44 \[ -\frac{\sqrt{-x^{2} + 1} x{\rm ln}\left (x + \sqrt{x^{2} + 1}\right )}{x^{2} - 1} - \frac{1}{2} \, \arcsin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(x + sqrt(x^2 + 1))/(-x^2 + 1)^(3/2),x, algorithm="giac")
[Out]