3.3.73 \(\int x \sin ^{-1}(x) \, dx\) [273]

Optimal. Leaf size=32 \[ \frac {1}{4} x \sqrt {1-x^2}-\frac {1}{4} \sin ^{-1}(x)+\frac {1}{2} x^2 \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4723, 327, 222} \begin {gather*} \frac {1}{2} x^2 \text {ArcSin}(x)-\frac {\text {ArcSin}(x)}{4}+\frac {1}{4} \sqrt {1-x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*ArcSin[x],x]

[Out]

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

Rule 222

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 4723

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.88 \begin {gather*} \frac {1}{4} \left (x \sqrt {1-x^2}+\left (-1+2 x^2\right ) \sin ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSin[x],x]

[Out]

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

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Maple [A]
time = 0.00, size = 25, normalized size = 0.78

method	result	size
default	\(-\frac {\arcsin \left (x \right )}{4}+\frac {x^{2} \arcsin \left (x \right )}{2}+\frac {x \sqrt {-x^{2}+1}}{4}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 3.01, size = 24, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, x^{2} \arcsin \left (x\right ) + \frac {1}{4} \, \sqrt {-x^{2} + 1} x - \frac {1}{4} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.17, size = 24, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right ) + \frac {1}{4} \, \sqrt {-x^{2} + 1} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.06, size = 24, normalized size = 0.75 \begin {gather*} \frac {x^{2} \operatorname {asin}{\left (x \right )}}{2} + \frac {x \sqrt {1 - x^{2}}}{4} - \frac {\operatorname {asin}{\left (x \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(x),x)

[Out]

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

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Giac [A]
time = 2.82, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 1\right )} \arcsin \left (x\right ) + \frac {1}{4} \, \sqrt {-x^{2} + 1} x + \frac {1}{4} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 24, normalized size = 0.75 \begin {gather*} \frac {x\,\sqrt {1-x^2}}{4}+\frac {\mathrm {asin}\left (x\right )\,\left (2\,x^2-1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asin(x),x)

[Out]

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

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