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]

-Log[1 - x^2]/2 + Log[ArcSin[x]]

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Rubi [A]  time = 0.0282553, antiderivative size = 16, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0233533, size = 16, normalized size = 1. \[ \log \left (\sin ^{-1}(x)\right )-\frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x^2]/2 + Log[ArcSin[x]]

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Maple [A]  time = 0.009, size = 17, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( x-1 \right ) }{2}}-{\frac{\ln \left ( x+1 \right ) }{2}}+\ln \left ( \arcsin \left ( x \right ) \right ) \end{align*}

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 = 1.45961, size = 16, normalized size = 1. \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (\arcsin \left (x\right )\right ) \end{align*}

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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Fricas [A]  time = 2.08817, size = 50, normalized size = 3.12 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \left (x\right )\right ) \end{align*}

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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Sympy [A]  time = 0.235039, size = 12, normalized size = 0.75 \begin{align*} - \frac{\log{\left (x^{2} - 1 \right )}}{2} + \log{\left (\operatorname{asin}{\left (x \right )} \right )} \end{align*}

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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Giac [A]  time = 1.14248, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \log \left ({\left | \arcsin \left (x\right ) \right |}\right ) \end{align*}

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)))