3.2.3 \(\int \frac {x^2 \text {ArcSin}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [103]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 = {4795, 4737, 30} \begin {gather*} \frac {\text {ArcSin}(a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{2 a^2}+\frac {x^2}{4 a} \end {gather*}

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 30

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

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

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*}

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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.86 \begin {gather*} \frac {a^2 x^2-2 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)+\text {ArcSin}(a x)^2}{4 a^3} \end {gather*}

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 = 0.08, size = 40, normalized size = 0.80

method	result	size
default	\(\frac {-2 a x \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}+\arcsin \left (a x \right )^{2}}{4 a^{3}}\)	\(40\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.48, size = 56, normalized size = 1.12 \begin {gather*} \frac {1}{4} \, a {\left (\frac {x^{2}}{a^{2}} - \frac {\arcsin \left (a x\right )^{2}}{a^{4}}\right )} - \frac {1}{2} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} \arcsin \left (a x\right ) \end {gather*}

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

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Fricas [A]
time = 1.68, size = 39, normalized size = 0.78 \begin {gather*} \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 {gather*}

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 = 0.28, size = 42, normalized size = 0.84 \begin {gather*} \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 {gather*}

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 = 0.41, size = 53, normalized size = 1.06 \begin {gather*} -\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 {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2\,\mathrm {asin}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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