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

Optimal. Leaf size=42 \[ \frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4643, 4641} \[ \frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^2/Sqrt[c - a^2*c*x^2],x]

[Out]

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

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 4643

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 42, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^2/Sqrt[c - a^2*c*x^2],x]

[Out]

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

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{2}}{a^{2} c x^{2} - c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^2/(a^2*c*x^2 - c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} c x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 52, normalized size = 1.24 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{3}}{3 a c \left (a^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.28, size = 14, normalized size = 0.33 \[ \frac {\arcsin \left (a x\right )^{3}}{3 \, a \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arcsin(a*x)^3/(a*sqrt(c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(a*x)**2/sqrt(-c*(a*x - 1)*(a*x + 1)), x)

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