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]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ a \sqrt {1-\frac {x^2}{a^2}}+x \csc ^{-1}\left (\frac {a}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[a/x],x]

[Out]

a*Sqrt[1 - x^2/a^2] + x*ArcCsc[a/x]

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fricas [A]  time = 0.92, size = 26, normalized size = 1.04 \[ x \operatorname {arccsc}\left (\frac {a}{x}\right ) + x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} \]

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)

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giac [A]  time = 0.13, size = 26, normalized size = 1.04 \[ a {\left (\frac {x \arcsin \left (\frac {x}{a}\right )}{a} + \sqrt {-\frac {x^{2}}{a^{2}} + 1}\right )} \]

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))

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maple [B]  time = 0.06, size = 52, normalized size = 2.08 \[ -a \left (-\frac {\mathrm {arccsc}\left (\frac {a}{x}\right ) x}{a}-\frac {x^{2} \left (-1+\frac {a^{2}}{x^{2}}\right )}{\sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}\, a^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(a/x),x)

[Out]

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

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maxima [A]  time = 0.46, size = 23, normalized size = 0.92 \[ x \operatorname {arccsc}\left (\frac {a}{x}\right ) + a \sqrt {-\frac {x^{2}}{a^{2}} + 1} \]

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)

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mupad [B]  time = 0.10, size = 23, normalized size = 0.92 \[ a\,\sqrt {1-\frac {x^2}{a^2}}+x\,\mathrm {asin}\left (\frac {x}{a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x/a),x)

[Out]

a*(1 - x^2/a^2)^(1/2) + x*asin(x/a)

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sympy [A]  time = 0.14, size = 22, normalized size = 0.88 \[ \begin {cases} a \sqrt {1 - \frac {x^{2}}{a^{2}}} + x \operatorname {acsc}{\left (\frac {a}{x} \right )} & \text {for}\: a \neq 0 \\\tilde {\infty } x & \text {otherwise} \end {cases} \]

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))

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