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

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {4767, 8} \begin {gather*} x-\sqrt {1-x^2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*ArcSin[x])/Sqrt[1 - x^2],x]

[Out]

x - Sqrt[1 - x^2]*ArcSin[x]

Rule 8

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

Rule 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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]

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*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} x-\sqrt {1-x^2} \sin ^{-1}(x) \end {gather*}

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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Mathics [A]
time = 1.68, size = 15, normalized size = 0.88 \begin {gather*} x-\text {ArcSin}\left [x\right ] \sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - ArcSin[x] Sqrt[1 - x ^ 2]

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Maple [A]
time = 0.06, size = 16, normalized size = 0.94

method	result	size
default	\(x -\arcsin \left (x \right ) \sqrt {-x^{2}+1}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 15, normalized size = 0.88 \begin {gather*} -\sqrt {-x^{2} + 1} \arcsin \left (x\right ) + x \end {gather*}

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 = 0.34, size = 15, normalized size = 0.88 \begin {gather*} -\sqrt {-x^{2} + 1} \arcsin \left (x\right ) + x \end {gather*}

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 = 0.09, size = 12, normalized size = 0.71 \begin {gather*} x - \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )} \end {gather*}

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 = 0.00, size = 16, normalized size = 0.94 \begin {gather*} -\sqrt {-x^{2}+1} \arcsin x+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x) + x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x\,\mathrm {asin}\left (x\right )}{\sqrt {1-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*asin(x))/(1 - x^2)^(1/2),x)

[Out]

int((x*asin(x))/(1 - x^2)^(1/2), x)

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