3.5.68 \(\int \frac {1}{\sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}} \, dx\) [468]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 42, 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} \sqrt {\text {ArcSin}(a x)}}{a \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)}}{a \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 38, normalized size = 0.90

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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