3.8 \(\int \frac{\sin ^{-1}(a x)}{x^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sin ^{-1}(a x)}{2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01436, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4627, 264} \[ -\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sin ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]/x^3,x]

[Out]

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

Rule 4627

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

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}(a x)}{x^3} \, dx &=-\frac{\sin ^{-1}(a x)}{2 x^2}+\frac{1}{2} a \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sin ^{-1}(a x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0071846, size = 29, normalized size = 0.85 \[ -\frac{a x \sqrt{1-a^2 x^2}+\sin ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 38, normalized size = 1.1 \begin{align*}{a}^{2} \left ( -{\frac{\arcsin \left ( ax \right ) }{2\,{a}^{2}{x}^{2}}}-{\frac{1}{2\,ax}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)/x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.65813, size = 38, normalized size = 1.12 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} a}{2 \, x} - \frac{\arcsin \left (a x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.1906, size = 66, normalized size = 1.94 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} a x + \arcsin \left (a x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 2.11402, size = 51, normalized size = 1.5 \begin{align*} \frac{a \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right )}{2} - \frac{\operatorname{asin}{\left (a x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)/x**3,x)

[Out]

a*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/2 - asin(a*x)/(2*
x**2)

________________________________________________________________________________________

Giac [B]  time = 1.28193, size = 92, normalized size = 2.71 \begin{align*} \frac{1}{4} \,{\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 )} a - \frac{\arcsin \left (a x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^3,x, algorithm="giac")

[Out]

1/4*(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)))*a - 1/2*arcs
in(a*x)/x^2