3.1.11 \(\int \csc ^{-1}(\frac {a}{x}) \, dx\) [11]

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

[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 = {5373, 4715, 267} \begin {gather*} a \sqrt {1-\frac {x^2}{a^2}}+x \text {ArcSin}\left (\frac {x}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a/x],x]

[Out]

a*Sqrt[1 - x^2/a^2] + x*ArcSin[x/a]

Rule 267

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 4715

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 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*} \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 \begin {gather*} a \sqrt {1-\frac {x^2}{a^2}}+x \csc ^{-1}\left (\frac {a}{x}\right ) \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(23)=46\).
time = 0.49, size = 52, normalized size = 2.08

method	result	size
derivativedivides	\(-a \left (-\frac {x \,\mathrm {arccsc}\left (\frac {a}{x}\right )}{a}-\frac {x^{2} \left (\frac {a^{2}}{x^{2}}-1\right )}{\sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{2}}\right )\)	\(52\)
default	\(-a \left (-\frac {x \,\mathrm {arccsc}\left (\frac {a}{x}\right )}{a}-\frac {x^{2} \left (\frac {a^{2}}{x^{2}}-1\right )}{\sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{2}}\right )\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 23, normalized size = 0.92 \begin {gather*} x \operatorname {arccsc}\left (\frac {a}{x}\right ) + a \sqrt {-\frac {x^{2}}{a^{2}} + 1} \end {gather*}

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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Fricas [A]
time = 0.34, size = 26, normalized size = 1.04 \begin {gather*} x \operatorname {arccsc}\left (\frac {a}{x}\right ) + x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} \end {gather*}

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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Sympy [A]
time = 0.06, size = 22, normalized size = 0.88 \begin {gather*} \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 {gather*}

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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Giac [A]
time = 0.43, size = 26, normalized size = 1.04 \begin {gather*} a {\left (\frac {x \arcsin \left (\frac {x}{a}\right )}{a} + \sqrt {-\frac {x^{2}}{a^{2}} + 1}\right )} \end {gather*}

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

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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