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

Optimal. Leaf size=26 \[ \sqrt{x^2-1} \log \left (\sqrt{x^2-1}+x\right )-x \]

[Out]

-x + Sqrt[-1 + x^2]*Log[x + Sqrt[-1 + x^2]]

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Rubi [A]  time = 0.0589585, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \sqrt{x^2-1} \log \left (\sqrt{x^2-1}+x\right )-x \]

Antiderivative was successfully verified.

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

[Out]

-x + Sqrt[-1 + x^2]*Log[x + Sqrt[-1 + x^2]]

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Rubi in Sympy [A]  time = 4.03534, size = 20, normalized size = 0.77 \[ - x + \sqrt{x^{2} - 1} \log{\left (x + \sqrt{x^{2} - 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0177875, size = 26, normalized size = 1. \[ \sqrt{x^2-1} \log \left (\sqrt{x^2-1}+x\right )-x \]

Antiderivative was successfully verified.

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

[Out]

-x + Sqrt[-1 + x^2]*Log[x + Sqrt[-1 + x^2]]

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{x\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.58119, size = 30, normalized size = 1.15 \[ \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.546861, size = 30, normalized size = 1.15 \[ \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 58.6406, size = 20, normalized size = 0.77 \[ - x + \sqrt{x^{2} - 1} \log{\left (x + \sqrt{x^{2} - 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.199897, size = 30, normalized size = 1.15 \[ \sqrt{x^{2} - 1}{\rm ln}\left (x + \sqrt{x^{2} - 1}\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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