3.10 \(\int \frac{\sin ^{-1}(a x)}{x^5} \, dx\)

Optimal. Leaf size=58 \[ -\frac{a^3 \sqrt{1-a^2 x^2}}{6 x}-\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{\sin ^{-1}(a x)}{4 x^4} \]

[Out]

-(a*Sqrt[1 - a^2*x^2])/(12*x^3) - (a^3*Sqrt[1 - a^2*x^2])/(6*x) - ArcSin[a*x]/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0224819, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4627, 271, 264} \[ -\frac{a^3 \sqrt{1-a^2 x^2}}{6 x}-\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{\sin ^{-1}(a x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]/x^5,x]

[Out]

-(a*Sqrt[1 - a^2*x^2])/(12*x^3) - (a^3*Sqrt[1 - a^2*x^2])/(6*x) - ArcSin[a*x]/(4*x^4)

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 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 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^5} \, dx &=-\frac{\sin ^{-1}(a x)}{4 x^4}+\frac{1}{4} a \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{\sin ^{-1}(a x)}{4 x^4}+\frac{1}{6} a^3 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{a^3 \sqrt{1-a^2 x^2}}{6 x}-\frac{\sin ^{-1}(a x)}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0186953, size = 41, normalized size = 0.71 \[ -\frac{a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+1\right )+3 \sin ^{-1}(a x)}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]/x^5,x]

[Out]

-(a*x*Sqrt[1 - a^2*x^2]*(1 + 2*a^2*x^2) + 3*ArcSin[a*x])/(12*x^4)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 58, normalized size = 1. \begin{align*}{a}^{4} \left ( -{\frac{\arcsin \left ( ax \right ) }{4\,{a}^{4}{x}^{4}}}-{\frac{1}{12\,{a}^{3}{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{6\,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^5,x)

[Out]

a^4*(-1/4*arcsin(a*x)/a^4/x^4-1/12/a^3/x^3*(-a^2*x^2+1)^(1/2)-1/6/a/x*(-a^2*x^2+1)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.65557, size = 68, normalized size = 1.17 \begin{align*} -\frac{1}{12} \,{\left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x} + \frac{\sqrt{-a^{2} x^{2} + 1}}{x^{3}}\right )} a - \frac{\arcsin \left (a x\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^5,x, algorithm="maxima")

[Out]

-1/12*(2*sqrt(-a^2*x^2 + 1)*a^2/x + sqrt(-a^2*x^2 + 1)/x^3)*a - 1/4*arcsin(a*x)/x^4

________________________________________________________________________________________

Fricas [A]  time = 2.53501, size = 89, normalized size = 1.53 \begin{align*} -\frac{{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{-a^{2} x^{2} + 1} + 3 \, \arcsin \left (a x\right )}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^5,x, algorithm="fricas")

[Out]

-1/12*((2*a^3*x^3 + a*x)*sqrt(-a^2*x^2 + 1) + 3*arcsin(a*x))/x^4

________________________________________________________________________________________

Sympy [A]  time = 3.91116, size = 100, normalized size = 1.72 \begin{align*} \frac{a \left (\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right )}{4} - \frac{\operatorname{asin}{\left (a x \right )}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)/x**5,x)

[Out]

a*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a*
*2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/4 - asin(a*x)/(4*x**4)

________________________________________________________________________________________

Giac [B]  time = 1.35443, size = 176, normalized size = 3.03 \begin{align*} \frac{1}{96} \,{\left (\frac{{\left (a^{4} + \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} - \frac{\frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2}{\left | a \right |}}\right )} a - \frac{\arcsin \left (a x\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^5,x, algorithm="giac")

[Out]

1/96*((a^4 + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) - (9*
(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/x^3)/(a^2*abs(a)))*a - 1/4*arcsin(a*
x)/x^4