\(\int \frac {\csc ^{-1}(\frac {a}{x})}{x^3} \, dx\) [14]

Optimal result

Integrand size = 10, antiderivative size = 38 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=-\frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\arcsin \left (\frac {x}{a}\right )}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5373, 4723, 270} \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=-\frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\arcsin \left (\frac {x}{a}\right )}{2 x^2} \]

[In]

Int[ArcCsc[a/x]/x^3,x]

[Out]

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

Rule 5373

Int[ArcCsc[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcSin[a/c + b*(x^n/c)]^m, x] /;
FreeQ[{a, b, c, n, m}, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\arcsin \left (\frac {x}{a}\right )}{x^3} \, dx \\ & = -\frac {\arcsin \left (\frac {x}{a}\right )}{2 x^2}+\frac {\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx}{2 a} \\ & = -\frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\arcsin \left (\frac {x}{a}\right )}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=-\frac {x \sqrt {1-\frac {x^2}{a^2}}+a \csc ^{-1}\left (\frac {a}{x}\right )}{2 a x^2} \]

[In]

Integrate[ArcCsc[a/x]/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87

method	result	size
parts	\(-\frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{2 x^{2}}-\frac {\sqrt {1-\frac {x^{2}}{a^{2}}}}{2 a x}\)	\(33\)
derivativedivides	\(-\frac {\frac {a^{2} \operatorname {arccsc}\left (\frac {a}{x}\right )}{2 x^{2}}+\frac {x \left (\frac {a^{2}}{x^{2}}-1\right )}{2 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a}}{a^{2}}\)	\(54\)
default	\(-\frac {\frac {a^{2} \operatorname {arccsc}\left (\frac {a}{x}\right )}{2 x^{2}}+\frac {x \left (\frac {a^{2}}{x^{2}}-1\right )}{2 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a}}{a^{2}}\)	\(54\)

[In]

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

[Out]

-1/2*arccsc(a/x)/x^2-1/2*(1-x^2/a^2)^(1/2)/a/x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=-\frac {a^{2} \operatorname {arccsc}\left (\frac {a}{x}\right ) + x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{2 \, a^{2} x^{2}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=- \frac {\operatorname {acsc}{\left (\frac {a}{x} \right )}}{2 x^{2}} + \frac {\begin {cases} - \frac {\sqrt {\frac {a^{2}}{x^{2}} - 1}}{a} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {i \sqrt {- \frac {a^{2}}{x^{2}} + 1}}{a} & \text {otherwise} \end {cases}}{2 a} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=-\frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{2 \, x^{2}} - \frac {\sqrt {-\frac {x^{2}}{a^{2}} + 1}}{2 \, a x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=-\frac {a {\left (\frac {a + \sqrt {a^{2} - x^{2}}}{a^{2} x} - \frac {x}{{\left (a + \sqrt {a^{2} - x^{2}}\right )} a^{2}}\right )}}{4 \, {\left | a \right |}} - \frac {\arcsin \left (\frac {x}{a}\right )}{2 \, x^{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx=\int \frac {\mathrm {asin}\left (\frac {x}{a}\right )}{x^3} \,d x \]

[In]

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

[Out]

int(asin(x/a)/x^3, x)