3.225 \(\int \frac {1}{x \sqrt {1-\log ^2(x)}} \, dx\)

Optimal. Leaf size=3 \[ \sin ^{-1}(\log (x)) \]

[Out]

arcsin(ln(x))

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {216} \[ \sin ^{-1}(\log (x)) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 - Log[x]^2]),x]

[Out]

ArcSin[Log[x]]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {1-\log ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\log (x)\right )\\ &=\sin ^{-1}(\log (x))\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 3, normalized size = 1.00 \[ \sin ^{-1}(\log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 - Log[x]^2]),x]

[Out]

ArcSin[Log[x]]

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 20, normalized size = 6.67 \[ -2 \, \arctan \left (\frac {\sqrt {-\log \relax (x)^{2} + 1} - 1}{\log \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 3, normalized size = 1.00 \[ \arcsin \left (\log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(log(x))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 4, normalized size = 1.33 \[ \arcsin \left (\ln \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.98, size = 3, normalized size = 1.00 \[ \arcsin \left (\log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(log(x))

________________________________________________________________________________________

mupad [B]  time = 0.39, size = 3, normalized size = 1.00 \[ \mathrm {asin}\left (\ln \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1 - log(x)^2)^(1/2)),x)

[Out]

asin(log(x))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {- \left (\log {\relax (x )} - 1\right ) \left (\log {\relax (x )} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________