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

Optimal. Leaf size=13 \[ \frac {\sin ^{-1}(a x)^2}{2 a} \]

[Out]

1/2*arcsin(a*x)^2/a

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {4641} \[ \frac {\sin ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

ArcSin[a*x]^2/(2*a)

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {\sin ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]^2/(2*a)

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fricas [A]  time = 0.55, size = 11, normalized size = 0.85 \[ \frac {\arcsin \left (a x\right )^{2}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(a*x)^2/a

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giac [A]  time = 0.36, size = 11, normalized size = 0.85 \[ \frac {\arcsin \left (a x\right )^{2}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(a*x)^2/a

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maple [A]  time = 0.01, size = 12, normalized size = 0.92 \[ \frac {\arcsin \left (a x \right )^{2}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(a*x)^2/a

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maxima [A]  time = 0.55, size = 11, normalized size = 0.85 \[ \frac {\arcsin \left (a x\right )^{2}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(a*x)^2/a

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mupad [B]  time = 0.14, size = 11, normalized size = 0.85 \[ \frac {{\mathrm {asin}\left (a\,x\right )}^2}{2\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)/(1 - a^2*x^2)^(1/2),x)

[Out]

asin(a*x)^2/(2*a)

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sympy [A]  time = 0.33, size = 10, normalized size = 0.77 \[ \begin {cases} \frac {\operatorname {asin}^{2}{\left (a x \right )}}{2 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((asin(a*x)**2/(2*a), Ne(a, 0)), (0, True))

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