3.5.44 \(\int \frac {\sqrt {\text {ArcSin}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) [444]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {4737} \begin {gather*} \frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^{3/2}}{3 a \sqrt {c-a^2 c x^2}} \end {gather*}

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 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]

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 = 0.03, size = 44, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^{3/2}}{3 a \sqrt {c-a^2 c x^2}} \end {gather*}

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])

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 38, normalized size = 0.86

method	result	size
default	\(\frac {2 \arcsin \left (a x \right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{3 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}\)	\(38\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\operatorname {asin}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________