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

Optimal. Leaf size=44 \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0752011, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0465079, size = 44, normalized size = 1. \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 38, normalized size = 0.9 \begin{align*}{\frac{2}{3\,a} \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{asin}{\left (a x \right )}}}{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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