3.668 \(\int \frac {x^2 \sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx\)

Optimal. Leaf size=73 \[ -\frac {3 x^2}{8}-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {3}{4} x^2 \sin ^{-1}(x)^2+\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3}{8} \sin ^{-1}(x)^2 \]

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Rubi [A]  time = 0.15, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {4707, 4641, 4627, 30} \[ -\frac {3 x^2}{8}-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {3}{4} x^2 \sin ^{-1}(x)^2+\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3}{8} \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[(x^2*ArcSin[x]^3)/Sqrt[1 - x^2],x]

[Out]

(-3*x^2)/8 + (3*x*Sqrt[1 - x^2]*ArcSin[x])/4 - (3*ArcSin[x]^2)/8 + (3*x^2*ArcSin[x]^2)/4 - (x*Sqrt[1 - x^2]*Ar
cSin[x]^3)/2 + ArcSin[x]^4/8

Rule 30

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

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4641

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

Rule 4707

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

Rubi steps

\begin {align*} \int \frac {x^2 \sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx &=-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{2} \int \frac {\sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx+\frac {3}{2} \int x \sin ^{-1}(x)^2 \, dx\\ &=\frac {3}{4} x^2 \sin ^{-1}(x)^2-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3}{2} \int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {3}{4} x^2 \sin ^{-1}(x)^2-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3 \int x \, dx}{4}-\frac {3}{4} \int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3 x^2}{8}+\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)-\frac {3}{8} \sin ^{-1}(x)^2+\frac {3}{4} x^2 \sin ^{-1}(x)^2-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{8} \sin ^{-1}(x)^4\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 60, normalized size = 0.82 \[ \frac {1}{8} \left (-3 x^2-4 x \sqrt {1-x^2} \sin ^{-1}(x)^3+\left (6 x^2-3\right ) \sin ^{-1}(x)^2+6 x \sqrt {1-x^2} \sin ^{-1}(x)+\sin ^{-1}(x)^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*ArcSin[x]^3)/Sqrt[1 - x^2],x]

[Out]

(-3*x^2 + 6*x*Sqrt[1 - x^2]*ArcSin[x] + (-3 + 6*x^2)*ArcSin[x]^2 - 4*x*Sqrt[1 - x^2]*ArcSin[x]^3 + ArcSin[x]^4
)/8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(x^2*ArcSin[x]^3)/Sqrt[1 - x^2],x]

[Out]

Could not integrate

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fricas [A]  time = 0.95, size = 49, normalized size = 0.67 \[ \frac {1}{8} \, \arcsin \relax (x)^{4} + \frac {3}{8} \, {\left (2 \, x^{2} - 1\right )} \arcsin \relax (x)^{2} - \frac {3}{8} \, x^{2} - \frac {1}{4} \, {\left (2 \, x \arcsin \relax (x)^{3} - 3 \, x \arcsin \relax (x)\right )} \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(x)^3/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/8*arcsin(x)^4 + 3/8*(2*x^2 - 1)*arcsin(x)^2 - 3/8*x^2 - 1/4*(2*x*arcsin(x)^3 - 3*x*arcsin(x))*sqrt(-x^2 + 1)

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giac [A]  time = 0.97, size = 60, normalized size = 0.82 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \relax (x)^{3} + \frac {1}{8} \, \arcsin \relax (x)^{4} + \frac {3}{4} \, {\left (x^{2} - 1\right )} \arcsin \relax (x)^{2} + \frac {3}{4} \, \sqrt {-x^{2} + 1} x \arcsin \relax (x) - \frac {3}{8} \, x^{2} + \frac {3}{8} \, \arcsin \relax (x)^{2} + \frac {3}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(x)^3/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-x^2 + 1)*x*arcsin(x)^3 + 1/8*arcsin(x)^4 + 3/4*(x^2 - 1)*arcsin(x)^2 + 3/4*sqrt(-x^2 + 1)*x*arcsin(
x) - 3/8*x^2 + 3/8*arcsin(x)^2 + 3/16

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maple [A]  time = 0.32, size = 69, normalized size = 0.95

method	result	size
default	\(\frac {\arcsin \relax (x )^{3} \left (-\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )\right )}{2}+\frac {3 \arcsin \relax (x )^{2} \left (x^{2}-1\right )}{4}+\frac {3 \arcsin \relax (x ) \left (\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )\right )}{4}-\frac {3 \arcsin \relax (x )^{2}}{8}-\frac {3 x^{2}}{8}-\frac {3 \arcsin \relax (x )^{4}}{8}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(x)^3/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*arcsin(x)^3*(-(-x^2+1)^(1/2)*x+arcsin(x))+3/4*arcsin(x)^2*(x^2-1)+3/4*arcsin(x)*((-x^2+1)^(1/2)*x+arcsin(x
))-3/8*arcsin(x)^2-3/8*x^2-3/8*arcsin(x)^4

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arcsin \relax (x)^{3}}{\sqrt {-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(x)^3/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2*arcsin(x)^3/sqrt(-x^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {asin}\relax (x)}^3}{\sqrt {1-x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*asin(x)^3)/(1 - x^2)^(1/2),x)

[Out]

int((x^2*asin(x)^3)/(1 - x^2)^(1/2), x)

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sympy [A]  time = 1.32, size = 66, normalized size = 0.90 \[ \frac {3 x^{2} \operatorname {asin}^{2}{\relax (x )}}{4} - \frac {3 x^{2}}{8} - \frac {x \sqrt {1 - x^{2}} \operatorname {asin}^{3}{\relax (x )}}{2} + \frac {3 x \sqrt {1 - x^{2}} \operatorname {asin}{\relax (x )}}{4} + \frac {\operatorname {asin}^{4}{\relax (x )}}{8} - \frac {3 \operatorname {asin}^{2}{\relax (x )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*asin(x)**3/(-x**2+1)**(1/2),x)

[Out]

3*x**2*asin(x)**2/4 - 3*x**2/8 - x*sqrt(1 - x**2)*asin(x)**3/2 + 3*x*sqrt(1 - x**2)*asin(x)/4 + asin(x)**4/8 -
3*asin(x)**2/8

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