3.474 \(\int \frac {1}{\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \, dx\)

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

[Out]

-2*(-a^2*x^2+1)^(1/2)/a/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ -\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 42, normalized size = 1.00 \[ -\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 48, normalized size = 1.14 \[ \frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} - a c\right )} \sqrt {\arcsin \left (a x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 38, normalized size = 0.90 \[ -\frac {2 \sqrt {-a^{2} x^{2}+1}}{\sqrt {\arcsin \left (a x \right )}\, a \sqrt {-c \left (a^{2} x^{2}-1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(asin(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2)),x)

[Out]

int(1/(asin(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*c*x**2+c)**(1/2)/asin(a*x)**(3/2),x)

[Out]

Integral(1/(sqrt(-c*(a*x - 1)*(a*x + 1))*asin(a*x)**(3/2)), x)

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