3.4.49 \(\int x \sin ^{-1}(x^2) \, dx\) [349]

Optimal. Leaf size=27 \[ \frac {\sqrt {1-x^4}}{2}+\frac {1}{2} x^2 \sin ^{-1}\left (x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6847, 4715, 267} \begin {gather*} \frac {\sqrt {1-x^4}}{2}+\frac {1}{2} x^2 \sin ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*ArcSin[x^2],x]

[Out]

Sqrt[1 - x^4]/2 + (x^2*ArcSin[x^2])/2

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4715

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

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{2} \left (\sqrt {1-x^4}+x^2 \sin ^{-1}\left (x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSin[x^2],x]

[Out]

(Sqrt[1 - x^4] + x^2*ArcSin[x^2])/2

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Mathics [A]
time = 1.95, size = 21, normalized size = 0.78 \begin {gather*} \frac {x^2 \text {ArcSin}\left [x^2\right ]}{2}+\frac {\sqrt {1-x^4}}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*ArcSin[x^2],x]')

[Out]

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

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Maple [A]
time = 0.00, size = 22, normalized size = 0.81

method	result	size
derivativedivides	\(\frac {x^{2} \arcsin \left (x^{2}\right )}{2}+\frac {\sqrt {-x^{4}+1}}{2}\)	\(22\)
default	\(\frac {x^{2} \arcsin \left (x^{2}\right )}{2}+\frac {\sqrt {-x^{4}+1}}{2}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsin(x^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac {1}{2} \, \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac {1}{2} \, \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.09, size = 19, normalized size = 0.70 \begin {gather*} \frac {x^{2} \operatorname {asin}{\left (x^{2} \right )}}{2} + \frac {\sqrt {1 - x^{4}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(x**2),x)

[Out]

x**2*asin(x**2)/2 + sqrt(1 - x**4)/2

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Giac [A]
time = 0.00, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{2} \arcsin \left (x^{2}\right )+\sqrt {1-x^{4}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x^2),x)

[Out]

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

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Mupad [B]
time = 0.28, size = 21, normalized size = 0.78 \begin {gather*} \frac {x^2\,\mathrm {asin}\left (x^2\right )}{2}+\frac {\sqrt {1-x^4}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asin(x^2),x)

[Out]

(x^2*asin(x^2))/2 + (1 - x^4)^(1/2)/2

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