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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.02, 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]

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

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fricas [A]  time = 0.85, size = 37, normalized size = 0.64 \[ -\frac {{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {-a^{2} x^{2} + 1} + 3 \, \arcsin \left (a x\right )}{12 \, x^{4}} \]

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

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giac [B]  time = 0.18, size = 130, normalized size = 2.24 \[ \frac {1}{96} \, {\left (\frac {{\left (a^{4} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2} {\left | a \right |}}\right )} a - \frac {\arcsin \left (a x\right )}{4 \, x^{4}} \]

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

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maple [A]  time = 0.00, size = 58, normalized size = 1.00 \[ a^{4} \left (-\frac {\arcsin \left (a x \right )}{4 a^{4} x^{4}}-\frac {\sqrt {-a^{2} x^{2}+1}}{12 a^{3} x^{3}}-\frac {\sqrt {-a^{2} x^{2}+1}}{6 a x}\right ) \]

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))

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maxima [A]  time = 0.43, size = 50, normalized size = 0.86 \[ -\frac {1}{12} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} a - \frac {\arcsin \left (a x\right )}{4 \, x^{4}} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {asin}\left (a\,x\right )}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)/x^5,x)

[Out]

int(asin(a*x)/x^5, x)

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sympy [A]  time = 1.73, size = 100, normalized size = 1.72 \[ \frac {a \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{4} - \frac {\operatorname {asin}{\left (a x \right )}}{4 x^{4}} \]

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)

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