3.7.59 \(\int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\) [659]

Optimal. Leaf size=34 \[ \frac {x^2}{4}-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} \sin ^{-1}(x)^2 \]

[Out]

1/4*x^2+1/4*arcsin(x)^2-1/2*x*arcsin(x)*(-x^2+1)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4795, 4737, 30} \begin {gather*} -\frac {1}{2} \sqrt {1-x^2} x \text {ArcSin}(x)+\frac {\text {ArcSin}(x)^2}{4}+\frac {x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*ArcSin[x])/Sqrt[1 - x^2],x]

[Out]

x^2/4 - (x*Sqrt[1 - x^2]*ArcSin[x])/2 + ArcSin[x]^2/4

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4737

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

Rule 4795

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

Rubi steps

\begin {align*} \int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx &=-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^2}{4}-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} \sin ^{-1}(x)^2\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.82 \begin {gather*} \frac {1}{4} \left (x^2-2 x \sqrt {1-x^2} \sin ^{-1}(x)+\sin ^{-1}(x)^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2 - 2*x*Sqrt[1 - x^2]*ArcSin[x] + ArcSin[x]^2)/4

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Maple [A]
time = 0.10, size = 32, normalized size = 0.94

method	result	size
default	\(\frac {\arcsin \left (x \right ) \left (-x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{2}-\frac {\arcsin \left (x \right )^{2}}{4}+\frac {x^{2}}{4}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(x)*(-x*(-x^2+1)^(1/2)+arcsin(x))-1/4*arcsin(x)^2+1/4*x^2

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Maxima [A]
time = 1.01, size = 32, normalized size = 0.94 \begin {gather*} \frac {1}{4} \, x^{2} - \frac {1}{2} \, {\left (\sqrt {-x^{2} + 1} x - \arcsin \left (x\right )\right )} \arcsin \left (x\right ) - \frac {1}{4} \, \arcsin \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2 - 1/2*(sqrt(-x^2 + 1)*x - arcsin(x))*arcsin(x) - 1/4*arcsin(x)^2

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Fricas [A]
time = 0.50, size = 26, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) + \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^2 + 1)*x*arcsin(x) + 1/4*x^2 + 1/4*arcsin(x)^2

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Sympy [A]
time = 0.10, size = 26, normalized size = 0.76 \begin {gather*} \frac {x^{2}}{4} - \frac {x \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{2} + \frac {\operatorname {asin}^{2}{\left (x \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/4 - x*sqrt(1 - x**2)*asin(x)/2 + asin(x)**2/4

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Giac [A]
time = 1.03, size = 27, normalized size = 0.79 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) + \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} - \frac {1}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^2 + 1)*x*arcsin(x) + 1/4*x^2 + 1/4*arcsin(x)^2 - 1/8

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^2\,\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^2*asin(x))/(1 - x^2)^(1/2),x)

[Out]

int((x^2*asin(x))/(1 - x^2)^(1/2), x)

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