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

Optimal. Leaf size=89 \[ \frac{x \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{\sin ^{-1}(a x)}{4 a^3}+\frac{x^2 \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.137485, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4707, 4641, 4627, 321, 216} \[ \frac{x \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{\sin ^{-1}(a x)}{4 a^3}+\frac{x^2 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 - a^2*x^2])/(4*a^2) - ArcSin[a*x]/(4*a^3) + (x^2*ArcSin[a*x])/(2*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*
x]^2)/(2*a^2) + ArcSin[a*x]^3/(6*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 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 321

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 216

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]

Rubi steps

\begin{align*} \int \frac{x^2 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\int \frac{\sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{2 a^2}+\frac{\int x \sin ^{-1}(a x) \, dx}{a}\\ &=\frac{x^2 \sin ^{-1}(a x)}{2 a}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{1}{2} \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x \sqrt{1-a^2 x^2}}{4 a^2}+\frac{x^2 \sin ^{-1}(a x)}{2 a}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{4 a^2}\\ &=\frac{x \sqrt{1-a^2 x^2}}{4 a^2}-\frac{\sin ^{-1}(a x)}{4 a^3}+\frac{x^2 \sin ^{-1}(a x)}{2 a}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}\\ \end{align*}

Mathematica [A]  time = 0.024274, size = 73, normalized size = 0.82 \[ \frac{3 a x \sqrt{1-a^2 x^2}-6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+\left (6 a^2 x^2-3\right ) \sin ^{-1}(a x)+2 \sin ^{-1}(a x)^3}{12 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*x*Sqrt[1 - a^2*x^2] + (-3 + 6*a^2*x^2)*ArcSin[a*x] - 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + 2*ArcSin[a*x
]^3)/(12*a^3)

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Maple [A]  time = 0.062, size = 71, normalized size = 0.8 \begin{align*}{\frac{1}{12\,{a}^{3}} \left ( -6\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}xa+6\,{a}^{2}{x}^{2}\arcsin \left ( ax \right ) +2\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\,ax\sqrt{-{a}^{2}{x}^{2}+1}-3\,\arcsin \left ( ax \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(-6*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*x*a+6*a^2*x^2*arcsin(a*x)+2*arcsin(a*x)^3+3*a*x*(-a^2*x^2+1)^(1/2)-3
*arcsin(a*x))/a^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \arcsin \left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73193, size = 150, normalized size = 1.69 \begin{align*} \frac{2 \, \arcsin \left (a x\right )^{3} + 3 \,{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) - 3 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{12 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*arcsin(a*x)^3 + 3*(2*a^2*x^2 - 1)*arcsin(a*x) - 3*sqrt(-a^2*x^2 + 1)*(2*a*x*arcsin(a*x)^2 - a*x))/a^3

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Sympy [A]  time = 1.50168, size = 78, normalized size = 0.88 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asin}{\left (a x \right )}}{2 a} - \frac{x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{2 a^{2}} + \frac{x \sqrt{- a^{2} x^{2} + 1}}{4 a^{2}} + \frac{\operatorname{asin}^{3}{\left (a x \right )}}{6 a^{3}} - \frac{\operatorname{asin}{\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)**2/(-a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((x**2*asin(a*x)/(2*a) - x*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(2*a**2) + x*sqrt(-a**2*x**2 + 1)/(4*a**
2) + asin(a*x)**3/(6*a**3) - asin(a*x)/(4*a**3), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.24809, size = 109, normalized size = 1.22 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{2 \, a^{2}} + \frac{\arcsin \left (a x\right )^{3}}{6 \, a^{3}} + \frac{\sqrt{-a^{2} x^{2} + 1} x}{4 \, a^{2}} + \frac{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{2 \, a^{3}} + \frac{\arcsin \left (a x\right )}{4 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^2/a^2 + 1/6*arcsin(a*x)^3/a^3 + 1/4*sqrt(-a^2*x^2 + 1)*x/a^2 + 1/2*(a^2*
x^2 - 1)*arcsin(a*x)/a^3 + 1/4*arcsin(a*x)/a^3