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

Optimal. Leaf size=39 \[ -\frac{\sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2} \log (x)}{x}-\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0483891, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2335, 277, 216} \[ -\frac{\sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2} \log (x)}{x}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x^2*Sqrt[1 - x^2]),x]

[Out]

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

Rule 2335

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n]))/(d*f*(m + 1)), x] - Dist[(b*n)/(d*(m + 1)), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, 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{\log (x)}{x^2 \sqrt{1-x^2}} \, dx &=-\frac{\sqrt{1-x^2} \log (x)}{x}+\int \frac{\sqrt{1-x^2}}{x^2} \, dx\\ &=-\frac{\sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2} \log (x)}{x}-\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{x}-\sin ^{-1}(x)-\frac{\sqrt{1-x^2} \log (x)}{x}\\ \end{align*}

Mathematica [A]  time = 0.0308564, size = 25, normalized size = 0.64 \[ -\frac{\sqrt{1-x^2} (\log (x)+1)}{x}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/(x^2*Sqrt[1 - x^2]),x]

[Out]

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

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Maple [A]  time = 0.033, size = 35, normalized size = 0.9 \begin{align*} -\arcsin \left ( x \right ) +{\frac{1}{x} \left ( -\ln \left ( x \right ) \sqrt{-{x}^{2}+1}-\sqrt{-{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.43639, size = 47, normalized size = 1.21 \begin{align*} -\frac{\sqrt{-x^{2} + 1} \log \left (x\right )}{x} - \frac{\sqrt{-x^{2} + 1}}{x} - \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.08141, size = 95, normalized size = 2.44 \begin{align*} \frac{2 \, x \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - \sqrt{-x^{2} + 1}{\left (\log \left (x\right ) + 1\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (x \right )}}{x^{2} \sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(x)/(x**2*sqrt(-(x - 1)*(x + 1))), x)

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Giac [B]  time = 1.09174, size = 99, normalized size = 2.54 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )} \log \left (x\right ) + \frac{x}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{\sqrt{-x^{2} + 1} - 1}{2 \, x} - \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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