3.3.98 \(\int \frac {\text {ArcSin}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx\) [298]

Optimal. Leaf size=42 \[ \frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^4}{4 a \sqrt {c-a^2 c x^2}} \]

[Out]

1/4*arcsin(a*x)^4*(-a^2*x^2+1)^(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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {4737} \begin {gather*} \frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^4}{4 a \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^4)/(4*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 {\sin ^{-1}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {\sin ^{-1}(a x)^3}{\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)^4}{4 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 {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^4}{4 a \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^4)/(4*a*Sqrt[c - a^2*c*x^2])

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Maple [A]
time = 0.09, size = 52, normalized size = 1.24

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4}}{4 a c \left (a^{2} x^{2}-1\right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-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)^4

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Maxima [A]
time = 0.50, size = 14, normalized size = 0.33 \begin {gather*} \frac {\arcsin \left (a x\right )^{4}}{4 \, a \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arcsin(a*x)^4/(a*sqrt(c))

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asin}^{3}{\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)**3/(-a**2*c*x**2+c)**(1/2),x)

[Out]

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

[Out]

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

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

[Out]

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

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