3.83 \(\int \sec ^{-1}(x) \, dx\)

Optimal. Leaf size=19 \[ x \sec ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]

[Out]

x*ArcSec[x] - ArcTanh[Sqrt[1 - x^(-2)]]

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Rubi [A]  time = 0.0280833, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2. \[ x \sec ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-\frac{1}{x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[ArcSec[x],x]

[Out]

x*ArcSec[x] - ArcTanh[Sqrt[1 - x^(-2)]]

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Rubi in Sympy [A]  time = 1.76474, size = 15, normalized size = 0.79 \[ x \operatorname{asec}{\left (x \right )} - \operatorname{atanh}{\left (\sqrt{1 - \frac{1}{x^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asec(x),x)

[Out]

x*asec(x) - atanh(sqrt(1 - 1/x**2))

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Mathematica [B]  time = 0.112068, size = 64, normalized size = 3.37 \[ x \sec ^{-1}(x)-\frac{\sqrt{x^2-1} \left (\log \left (\frac{x}{\sqrt{x^2-1}}+1\right )-\log \left (1-\frac{x}{\sqrt{x^2-1}}\right )\right )}{2 \sqrt{1-\frac{1}{x^2}} x} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSec[x],x]

[Out]

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

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Maple [A]  time = 0.004, size = 22, normalized size = 1.2 \[ x{\rm arcsec} \left (x\right )-\ln \left ( x+x\sqrt{1-{x}^{-2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsec(x),x)

[Out]

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

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Maxima [A]  time = 1.37341, size = 47, normalized size = 2.47 \[ x \operatorname{arcsec}\left (x\right ) - \frac{1}{2} \, \log \left (\sqrt{-\frac{1}{x^{2}} + 1} + 1\right ) + \frac{1}{2} \, \log \left (-\sqrt{-\frac{1}{x^{2}} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.241886, size = 45, normalized size = 2.37 \[{\left (x - 2\right )} \operatorname{arcsec}\left (x\right ) + 4 \, \arctan \left (-x + \sqrt{x^{2} - 1}\right ) + \log \left (-x + \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x),x, algorithm="fricas")

[Out]

(x - 2)*arcsec(x) + 4*arctan(-x + sqrt(x^2 - 1)) + log(-x + sqrt(x^2 - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \operatorname{asec}{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asec(x),x)

[Out]

Integral(asec(x), x)

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GIAC/XCAS [A]  time = 0.231164, size = 34, normalized size = 1.79 \[ x \arccos \left (\frac{1}{x}\right ) + \frac{{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right )}{{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsec(x),x, algorithm="giac")

[Out]

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