3.1.27 \(\int \frac {\log (x)}{x^2 \sqrt {1-x^2}} \, dx\) [27]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {2373, 283, 222} \begin {gather*} -\text {ArcSin}(x)-\frac {\sqrt {1-x^2}}{x}-\frac {\sqrt {1-x^2} \log (x)}{x} \end {gather*}

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 222

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]

Rule 283

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 2373

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]

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*}

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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.64 \begin {gather*} -\sin ^{-1}(x)-\frac {\sqrt {1-x^2} (1+\log (x))}{x} \end {gather*}

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 = 0.03, size = 35, normalized size = 0.90

method	result	size
meijerg	\(-\arcsin \left (x \right )+\frac {-\ln \left (x \right ) \sqrt {-x^{2}+1}-\sqrt {-x^{2}+1}}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 4.06, size = 35, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {-x^{2} + 1} \log \left (x\right )}{x} - \frac {\sqrt {-x^{2} + 1}}{x} - \arcsin \left (x\right ) \end {gather*}

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 = 0.44, size = 39, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (x \right )}}{x^{2} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (35) = 70\).
time = 0.45, size = 73, normalized size = 1.87 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (x\right )}{x^2\,\sqrt {1-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(x^2*(1 - x^2)^(1/2)),x)

[Out]

int(log(x)/(x^2*(1 - x^2)^(1/2)), x)

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