3.15 \(\int \frac{\log \left (x+\sqrt{1+x^2}\right )}{\left (1-x^2\right )^{3/2}} \, dx\)

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]

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

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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]

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

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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]

x*log(x + sqrt(x**2 + 1))/sqrt(-x**2 + 1) - asin(x**2)/2

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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]

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

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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]

int(ln(x+(x^2+1)^(1/2))/(-x^2+1)^(3/2),x)

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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]

integrate(log(x + sqrt(x^2 + 1))/(-x^2 + 1)^(3/2), x)

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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]

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

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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]

Timed 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]

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