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

Optimal. Leaf size=50 \[ -\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 a^2}+\frac{\sin ^{-1}(a x)^2}{4 a^3}+\frac{x^2}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.0818071, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4707, 4641, 30} \[ -\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 a^2}+\frac{\sin ^{-1}(a x)^2}{4 a^3}+\frac{x^2}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 30

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

Rubi steps

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

Mathematica [A]  time = 0.0110443, size = 43, normalized size = 0.86 \[ \frac{a^2 x^2-2 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+\sin ^{-1}(a x)^2}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 40, normalized size = 0.8 \begin{align*}{\frac{1}{4\,{a}^{3}} \left ( -2\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa+{a}^{2}{x}^{2}+ \left ( \arcsin \left ( ax \right ) \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.58923, size = 101, normalized size = 2.02 \begin{align*} \frac{1}{4} \, a{\left (\frac{x^{2}}{a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )^{2}}{a^{4}}\right )} - \frac{1}{2} \,{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} \arcsin \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.04795, size = 100, normalized size = 2. \begin{align*} \frac{a^{2} x^{2} - 2 \, \sqrt{-a^{2} x^{2} + 1} a x \arcsin \left (a x\right ) + \arcsin \left (a x\right )^{2}}{4 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.931884, size = 42, normalized size = 0.84 \begin{align*} \begin{cases} \frac{x^{2}}{4 a} - \frac{x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{2 a^{2}} + \frac{\operatorname{asin}^{2}{\left (a x \right )}}{4 a^{3}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2/(4*a) - x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(2*a**2) + asin(a*x)**2/(4*a**3), Ne(a, 0)), (0, True
))

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Giac [A]  time = 1.38191, size = 72, normalized size = 1.44 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{2 \, a^{2}} + \frac{\arcsin \left (a x\right )^{2}}{4 \, a^{3}} + \frac{a^{2} x^{2} - 1}{4 \, a^{3}} + \frac{1}{8 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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