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 = 0.0293864, 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, Rules used = {5236, 29} \[ \sqrt{x^2-1} \sec ^{-1}(x)-\frac{x \log (x)}{\sqrt{x^2}} \]

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]

Rule 5236

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcSec[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[c^2*x^2]), Int[(d + e*x^2)^(p + 1)/
(x*Sqrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{x \sec ^{-1}(x)}{\sqrt{-1+x^2}} \, dx &=\sqrt{-1+x^2} \sec ^{-1}(x)-\frac{x \int \frac{1}{x} \, dx}{\sqrt{x^2}}\\ &=\sqrt{-1+x^2} \sec ^{-1}(x)-\frac{x \log (x)}{\sqrt{x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0418904, 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 = 0.355, size = 97, normalized size = 3.9 \begin{align*}{-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}}}} \end{align*}

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 = 0.97353, size = 20, normalized size = 0.8 \begin{align*} \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 - 1)*arcsec(x) - log(x)

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Fricas [A]  time = 2.17265, size = 46, normalized size = 1.84 \begin{align*} \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 - 1)*arcsec(x) - log(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{asec}{\left (x \right )}}{\sqrt{\left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*asec(x)/sqrt((x - 1)*(x + 1)), x)

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Giac [A]  time = 1.11712, size = 30, normalized size = 1.2 \begin{align*} \sqrt{x^{2} - 1} \arccos \left (\frac{1}{x}\right ) - \frac{\log \left ({\left | x \right |}\right )}{\mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 - 1)*arccos(1/x) - log(abs(x))/sgn(x)