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

Optimal. Leaf size=21 \[ \log (x)-\frac{\sqrt{1-x^2} \sin ^{-1}(x)}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0436064, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4681, 29} \[ \log (x)-\frac{\sqrt{1-x^2} \sin ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]/(x^2*Sqrt[1 - x^2]),x]

[Out]

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

Rule 4681

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0137963, size = 21, normalized size = 1. \[ \log (x)-\frac{\sqrt{1-x^2} \sin ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 20, normalized size = 1. \begin{align*} \ln \left ( x \right ) -{\frac{\arcsin \left ( x \right ) }{x}\sqrt{-{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.42703, size = 26, normalized size = 1.24 \begin{align*} -\frac{\sqrt{-x^{2} + 1} \arcsin \left (x\right )}{x} + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x)/x + log(x)

________________________________________________________________________________________

Fricas [A]  time = 2.15627, size = 55, normalized size = 2.62 \begin{align*} \frac{x \log \left (x\right ) - \sqrt{-x^{2} + 1} \arcsin \left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}{\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(asin(x)/x**2/(-x**2+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.08992, size = 54, normalized size = 2.57 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )} \arcsin \left (x\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(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)*arcsin(x) + log(abs(x))