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

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

[Out]

ArcSin[Log[x]]

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Rubi [A]  time = 0.0335502, antiderivative size = 3, normalized size of antiderivative = 1., 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 = 0.017991, size = 3, normalized size = 1. \[ \sin ^{-1}(\log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 - Log[x]^2]),x]

[Out]

ArcSin[Log[x]]

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Maple [A]  time = 0.008, size = 4, normalized size = 1.3 \begin{align*} \arcsin \left ( \ln \left ( x \right ) \right ) \end{align*}

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))

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Maxima [A]  time = 1.50694, size = 4, normalized size = 1.33 \begin{align*} \arcsin \left (\log \left (x\right )\right ) \end{align*}

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))

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Fricas [B]  time = 1.76992, size = 61, normalized size = 20.33 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-\log \left (x\right )^{2} + 1} - 1}{\log \left (x\right )}\right ) \end{align*}

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))

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

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)

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Giac [A]  time = 1.32127, size = 4, normalized size = 1.33 \begin{align*} \arcsin \left (\log \left (x\right )\right ) \end{align*}

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))