3.16 \(\int \sin ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=35 \[ \frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4619, 4677, 8} \[ \frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 35, normalized size = 1.00 \[ \frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^2,x]

[Out]

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

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fricas [A]  time = 0.50, size = 36, normalized size = 1.03 \[ \frac {a x \arcsin \left (a x\right )^{2} - 2 \, a x + 2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.12, size = 33, normalized size = 0.94 \[ x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 37, normalized size = 1.06 \[ \frac {a x \arcsin \left (a x \right )^{2}-2 a x +2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^2,x)

[Out]

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

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maxima [A]  time = 0.71, size = 33, normalized size = 0.94 \[ x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.14, size = 32, normalized size = 0.91 \[ x\,\left ({\mathrm {asin}\left (a\,x\right )}^2-2\right )+\frac {2\,\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^2,x)

[Out]

x*(asin(a*x)^2 - 2) + (2*asin(a*x)*(1 - a^2*x^2)^(1/2))/a

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sympy [A]  time = 0.18, size = 32, normalized size = 0.91 \[ \begin {cases} x \operatorname {asin}^{2}{\left (a x \right )} - 2 x + \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**2,x)

[Out]

Piecewise((x*asin(a*x)**2 - 2*x + 2*sqrt(-a**2*x**2 + 1)*asin(a*x)/a, Ne(a, 0)), (0, True))

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