### 3.349 $$\int x \sin ^{-1}(x^2) \, dx$$

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

[Out]

Sqrt[1 - x^4]/2 + (x^2*ArcSin[x^2])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0152655, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {6715, 4619, 261} $\frac{\sqrt{1-x^4}}{2}+\frac{1}{2} x^2 \sin ^{-1}\left (x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sqrt[1 - x^4]/2 + (x^2*ArcSin[x^2])/2

Rule 6715

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]

Rule 4619

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 261

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]

Rubi steps

\begin{align*} \int x \sin ^{-1}\left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{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} \operatorname{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*}

Mathematica [A]  time = 0.0044751, size = 24, normalized size = 0.89 $\frac{1}{2} \left (\sqrt{1-x^4}+x^2 \sin ^{-1}\left (x^2\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[1 - x^4] + x^2*ArcSin[x^2])/2

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 22, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}\arcsin \left ({x}^{2} \right ) }{2}}+{\frac{1}{2}\sqrt{-{x}^{4}+1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.41565, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 2.05932, size = 57, normalized size = 2.11 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.17522, size = 19, normalized size = 0.7 \begin{align*} \frac{x^{2} \operatorname{asin}{\left (x^{2} \right )}}{2} + \frac{\sqrt{1 - x^{4}}}{2} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.05492, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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