3.35 \(\int \frac{x \sec ^{-1}(x)}{\sqrt{-1+x^2}} \, dx\)

Optimal. Leaf size=25 \[ \sqrt{x^2-1} \sec ^{-1}(x)-\frac{x \log (x)}{\sqrt{x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0461073, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \sqrt{x^2-1} \sec ^{-1}(x)-\frac{x \log (x)}{\sqrt{x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.32509, size = 22, normalized size = 0.88 \[ - \frac{x \log{\left (x \right )}}{\sqrt{x^{2}}} + \sqrt{x^{2} - 1} \operatorname{asec}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0584276, size = 35, normalized size = 1.4 \[ \frac{\left (x^2-1\right ) \sec ^{-1}(x)-\sqrt{1-\frac{1}{x^2}} x \log (x)}{\sqrt{x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.465, size = 97, normalized size = 3.9 \[{-2\,ix{\rm arcsec} \left (x\right )\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{\frac{1}{\sqrt{{x}^{2}-1}}}}+{{\rm arcsec} \left (x\right ) \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{x\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ( \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) ^{2}+1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*I/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*arcsec(x)+1/(x^2-1)^(1/2)*(I*((x^2-1)/x
^2)^(1/2)*x+x^2-1)*arcsec(x)+1/(x^2-1)^(1/2)*((x^2-1)/x^2)^(1/2)*x*ln((1/x+I*(1-
1/x^2)^(1/2))^2+1)

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Maxima [A]  time = 1.47103, size = 20, normalized size = 0.8 \[ \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 - 1)*arcsec(x) - log(x)

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Fricas [A]  time = 0.245634, size = 20, normalized size = 0.8 \[ \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 - 1)*arcsec(x) - log(x)

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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(x*asec(x)/(x**2-1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21297, size = 30, normalized size = 1.2 \[ \sqrt{x^{2} - 1} \arccos \left (\frac{1}{x}\right ) - \frac{{\rm ln}\left ({\left | x \right |}\right )}{{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 - 1)*arccos(1/x) - ln(abs(x))/sign(x)