\(\int \frac {\arcsin (a x)}{x^3} \, dx\) [8]

Optimal result

Integrand size = 8, antiderivative size = 34 \[ \int \frac {\arcsin (a x)}{x^3} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\arcsin (a x)}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4723, 270} \[ \int \frac {\arcsin (a x)}{x^3} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\arcsin (a x)}{2 x^2} \]

[In]

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

[Out]

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

Rule 270

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 4723

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*} \text {integral}& = -\frac {\arcsin (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 {\arcsin (a x)}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {\arcsin (a x)}{x^3} \, dx=-\frac {a x \sqrt {1-a^2 x^2}+\arcsin (a x)}{2 x^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85

method	result	size
parts	\(-\frac {\arcsin \left (a x \right )}{2 x^{2}}-\frac {a \sqrt {-a^{2} x^{2}+1}}{2 x}\)	\(29\)
derivativedivides	\(a^{2} \left (-\frac {\arcsin \left (a x \right )}{2 a^{2} x^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{2 a x}\right )\)	\(38\)
default	\(a^{2} \left (-\frac {\arcsin \left (a x \right )}{2 a^{2} x^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{2 a x}\right )\)	\(38\)

[In]

int(arcsin(a*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \frac {\arcsin (a x)}{x^3} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} a x + \arcsin \left (a x\right )}{2 \, x^{2}} \]

[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] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {\arcsin (a x)}{x^3} \, dx=\frac {a \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{2} - \frac {\operatorname {asin}{\left (a x \right )}}{2 x^{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {\arcsin (a x)}{x^3} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} a}{2 \, x} - \frac {\arcsin \left (a x\right )}{2 \, x^{2}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {\arcsin (a x)}{x^3} \, dx=\frac {1}{4} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} a - \frac {\arcsin \left (a x\right )}{2 \, x^{2}} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)}{x^3} \, dx=\int \frac {\mathrm {asin}\left (a\,x\right )}{x^3} \,d x \]

[In]

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

[Out]

int(asin(a*x)/x^3, x)