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

Optimal. Leaf size=17 \[ x-\sqrt{1-x^2} \sin ^{-1}(x) \]

[Out]

x - Sqrt[1 - x^2]*ArcSin[x]

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Rubi [A]  time = 0.0284056, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4677, 8} \[ x-\sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcSin[x])/Sqrt[1 - x^2],x]

[Out]

x - Sqrt[1 - x^2]*ArcSin[x]

Rule 4677

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0053169, size = 17, normalized size = 1. \[ x-\sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x - Sqrt[1 - x^2]*ArcSin[x]

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Maple [A]  time = 0.044, size = 16, normalized size = 0.9 \begin{align*} x-\arcsin \left ( x \right ) \sqrt{-{x}^{2}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.43044, size = 20, normalized size = 1.18 \begin{align*} -\sqrt{-x^{2} + 1} \arcsin \left (x\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x) + x

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Fricas [A]  time = 2.47955, size = 42, normalized size = 2.47 \begin{align*} -\sqrt{-x^{2} + 1} \arcsin \left (x\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x) + x

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Sympy [A]  time = 0.187062, size = 12, normalized size = 0.71 \begin{align*} x - \sqrt{1 - x^{2}} \operatorname{asin}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - sqrt(1 - x**2)*asin(x)

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Giac [A]  time = 1.12276, size = 20, normalized size = 1.18 \begin{align*} -\sqrt{-x^{2} + 1} \arcsin \left (x\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x) + x