3.473 \(\int (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)}) \, dx\)

Optimal. Leaf size=16 \[ \log \left (\sin ^{-1}(x)\right )-\frac {1}{2} \log \left (1-x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {260, 4639} \[ \log \left (\sin ^{-1}(x)\right )-\frac {1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x^2]/2 + Log[ArcSin[x]]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4639

Int[1/(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[Log[a + b*ArcSin[c*x]]
/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 16, normalized size = 1.00 \[ \log \left (\sin ^{-1}(x)\right )-\frac {1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Log[1 - x^2] + Log[ArcSin[x]]

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fricas [A]  time = 0.42, size = 14, normalized size = 0.88 \[ -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x^2 - 1) + log(-arcsin(x))

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giac [A]  time = 1.88, size = 14, normalized size = 0.88 \[ -\frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \log \left ({\left | \arcsin \relax (x) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 17, normalized size = 1.06 \[ -\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+\ln \left (\arcsin \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 12, normalized size = 0.75 \[ -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (\arcsin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x^2 - 1) + log(arcsin(x))

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mupad [B]  time = 0.31, size = 12, normalized size = 0.75 \[ \ln \left (\mathrm {asin}\relax (x)\right )-\frac {\ln \left (x^2-1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(asin(x)*(1 - x^2)^(1/2)) - x/(x^2 - 1),x)

[Out]

log(asin(x)) - log(x^2 - 1)/2

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sympy [A]  time = 0.17, size = 12, normalized size = 0.75 \[ - \frac {\log {\left (x^{2} - 1 \right )}}{2} + \log {\left (\operatorname {asin}{\relax (x )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x**2 - 1)/2 + log(asin(x))

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