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

Optimal. Leaf size=32 \[ \frac{x \log \left (\sqrt{x^2-1}+x\right )}{\sqrt{x^2+1}}-\frac{1}{2} \cosh ^{-1}\left (x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0853702, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \log \left (\sqrt{x^2-1}+x\right )}{\sqrt{x^2+1}}-\frac{1}{2} \cosh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[Log[x + Sqrt[-1 + x^2]]/(1 + x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.87101, size = 27, normalized size = 0.84 \[ \frac{x \log{\left (x + \sqrt{x^{2} - 1} \right )}}{\sqrt{x^{2} + 1}} - \frac{\operatorname{acosh}{\left (x^{2} \right )}}{2} \]

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) - acosh(x**2)/2

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Mathematica [B]  time = 0.118346, size = 89, normalized size = 2.78 \[ \frac{4 x \log \left (\sqrt{x^2-1}+x\right )+\frac{\sqrt{x^2-1} \left (x^2+1\right ) \left (\log \left (1-\frac{x^2}{\sqrt{x^4-1}}\right )-\log \left (\frac{x^2}{\sqrt{x^4-1}}+1\right )\right )}{\sqrt{x^4-1}}}{4 \sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[Log[x + Sqrt[-1 + x^2]]/(1 + x^2)^(3/2),x]

[Out]

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

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Maple [F]  time = 0.016, 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 = 0.230715, size = 78, normalized size = 2.44 \[ \frac{2 \, \sqrt{x^{2} + 1} x \log \left (x + \sqrt{x^{2} - 1}\right ) +{\left (x^{2} + 1\right )} \log \left (-x^{2} + \sqrt{x^{2} + 1} \sqrt{x^{2} - 1}\right )}{2 \,{\left (x^{2} + 1\right )}} \]

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="fricas")

[Out]

1/2*(2*sqrt(x^2 + 1)*x*log(x + sqrt(x^2 - 1)) + (x^2 + 1)*log(-x^2 + sqrt(x^2 +
1)*sqrt(x^2 - 1)))/(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 = 0.20832, size = 49, normalized size = 1.53 \[ \frac{x{\rm ln}\left (x + \sqrt{x^{2} - 1}\right )}{\sqrt{x^{2} + 1}} + \frac{1}{2} \,{\rm ln}\left (x^{2} - \sqrt{x^{4} - 1}\right ) \]

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]

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