3.2.4 \(\int \frac {x \text {ArcSin}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [104]

Optimal. Leaf size=29 \[ \frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{a^2} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4767, 8} \begin {gather*} \frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

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

Rule 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac {\int 1 \, dx}{a}\\ &=\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(27)=54\).
time = 0.08, size = 62, normalized size = 2.14

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2} \arcsin \left (a x \right )-\arcsin \left (a x \right )+a x \sqrt {-a^{2} x^{2}+1}\right )}{a^{2} \left (a^{2} x^{2}-1\right )}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 27, normalized size = 0.93 \begin {gather*} \frac {x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.24, size = 26, normalized size = 0.90 \begin {gather*} \frac {a x - \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.21, size = 24, normalized size = 0.83 \begin {gather*} \begin {cases} \frac {x}{a} - \frac {\sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.40, size = 27, normalized size = 0.93 \begin {gather*} \frac {x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x\,\mathrm {asin}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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