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

Optimal. Leaf size=25 \[ a \sqrt{1-\frac{x^2}{a^2}}+x \sin ^{-1}\left (\frac{x}{a}\right ) \]

[Out]

a*Sqrt[1 - x^2/a^2] + x*ArcSin[x/a]

________________________________________________________________________________________

Rubi [A]  time = 0.0097707, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5265, 4619, 261} \[ a \sqrt{1-\frac{x^2}{a^2}}+x \sin ^{-1}\left (\frac{x}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a/x],x]

[Out]

a*Sqrt[1 - x^2/a^2] + x*ArcSin[x/a]

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 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0116146, size = 25, normalized size = 1. \[ a \sqrt{1-\frac{x^2}{a^2}}+x \csc ^{-1}\left (\frac{a}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[a/x],x]

[Out]

a*Sqrt[1 - x^2/a^2] + x*ArcCsc[a/x]

________________________________________________________________________________________

Maple [B]  time = 0.182, size = 52, normalized size = 2.1 \begin{align*} -a \left ( -{\frac{x}{a}{\rm arccsc} \left ({\frac{a}{x}}\right )}-{\frac{{x}^{2}}{{a}^{2}} \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)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00991, size = 31, normalized size = 1.24 \begin{align*} x \operatorname{arccsc}\left (\frac{a}{x}\right ) + a \sqrt{-\frac{x^{2}}{a^{2}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccsc(a/x) + a*sqrt(-x^2/a^2 + 1)

________________________________________________________________________________________

Fricas [A]  time = 2.97449, size = 55, normalized size = 2.2 \begin{align*} x \operatorname{arccsc}\left (\frac{a}{x}\right ) + x \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.221764, size = 22, normalized size = 0.88 \begin{align*} \begin{cases} a \sqrt{1 - \frac{x^{2}}{a^{2}}} + x \operatorname{acsc}{\left (\frac{a}{x} \right )} & \text{for}\: a \neq 0 \\\tilde{\infty } x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsc(a/x),x)

[Out]

Piecewise((a*sqrt(1 - x**2/a**2) + x*acsc(a/x), Ne(a, 0)), (zoo*x, True))

________________________________________________________________________________________

Giac [A]  time = 1.08226, size = 35, normalized size = 1.4 \begin{align*} a{\left (\frac{x \arcsin \left (\frac{x}{a}\right )}{a} + \sqrt{-\frac{x^{2}}{a^{2}} + 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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