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

Optimal. Leaf size=38 \[ -\frac{\sqrt{1-\frac{x^2}{a^2}}}{2 a x}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0221313, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5265, 4627, 264} \[ -\frac{\sqrt{1-\frac{x^2}{a^2}}}{2 a x}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5265

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]

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{\csc ^{-1}\left (\frac{a}{x}\right )}{x^3} \, dx &=\int \frac{\sin ^{-1}\left (\frac{x}{a}\right )}{x^3} \, dx\\ &=-\frac{\sin ^{-1}\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{\sin ^{-1}\left (\frac{x}{a}\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0266303, size = 35, normalized size = 0.92 \[ -\frac{x \sqrt{1-\frac{x^2}{a^2}}+a \csc ^{-1}\left (\frac{a}{x}\right )}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.186, size = 54, normalized size = 1.4 \begin{align*} -{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}}{2\,{x}^{2}}{\rm arccsc} \left ({\frac{a}{x}}\right )}+{\frac{x}{2\,a} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.2053, size = 84, normalized size = 2.21 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsc}{\left (\frac{a}{x} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acsc(a/x)/x**3, x)

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Giac [A]  time = 1.11512, size = 82, normalized size = 2.16 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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