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

Optimal. Leaf size=28 \[ a \log (x)-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4681, 29} \[ a \log (x)-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 4681

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 28, normalized size = 1.00 \[ a \log (x)-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.97, size = 28, normalized size = 1.00 \[ \frac {a x \log \relax (x) - \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.36, size = 67, normalized size = 2.39 \[ \frac {1}{2} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \arcsin \left (a x\right ) + a \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a)))*arcsin(a*x)
+ a*log(abs(x))

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maple [A]  time = 0.09, size = 32, normalized size = 1.14 \[ -\frac {-\ln \left (a x \right ) a x +\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 26, normalized size = 0.93 \[ a \log \relax (x) - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {asin}\left (a\,x\right )}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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