3.1.10 \(\int x \csc ^{-1}(\frac {a}{x}) \, dx\) [10]

Optimal. Leaf size=47 \[ \frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{4} a^2 \text {ArcSin}\left (\frac {x}{a}\right )+\frac {1}{2} x^2 \text {ArcSin}\left (\frac {x}{a}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5373, 4723, 327, 222} \begin {gather*} -\frac {1}{4} a^2 \text {ArcSin}\left (\frac {x}{a}\right )+\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \text {ArcSin}\left (\frac {x}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*ArcCsc[a/x],x]

[Out]

(a*x*Sqrt[1 - x^2/a^2])/4 - (a^2*ArcSin[x/a])/4 + (x^2*ArcSin[x/a])/2

Rule 222

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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*} \int x \csc ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int x \sin ^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=\frac {1}{2} x^2 \sin ^{-1}\left (\frac {x}{a}\right )-\frac {\int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{2 a}\\ &=\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \sin ^{-1}\left (\frac {x}{a}\right )-\frac {1}{4} a \int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx\\ &=\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{4} a^2 \sin ^{-1}\left (\frac {x}{a}\right )+\frac {1}{2} x^2 \sin ^{-1}\left (\frac {x}{a}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 44, normalized size = 0.94 \begin {gather*} \frac {1}{4} \left (a x \sqrt {1-\frac {x^2}{a^2}}+2 x^2 \csc ^{-1}\left (\frac {a}{x}\right )-a^2 \text {ArcSin}\left (\frac {x}{a}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*ArcCsc[a/x],x]

[Out]

(a*x*Sqrt[1 - x^2/a^2] + 2*x^2*ArcCsc[a/x] - a^2*ArcSin[x/a])/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(39)=78\).
time = 0.49, size = 90, normalized size = 1.91

method	result	size
derivativedivides	\(-a^{2} \left (-\frac {x^{2} \mathrm {arccsc}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (-\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}+\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\)	\(90\)
default	\(-a^{2} \left (-\frac {x^{2} \mathrm {arccsc}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (-\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}+\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*(-1/2*x^2/a^2*arccsc(a/x)-1/4*(a^2/x^2-1)^(1/2)*(-arctan(1/(a^2/x^2-1)^(1/2))/x^2*a^2+(a^2/x^2-1)^(1/2))/
((a^2/x^2-1)*x^2/a^2)^(1/2)*x^3/a^3)

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Maxima [A]
time = 0.46, size = 46, normalized size = 0.98 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\frac {a}{x}\right ) - \frac {a^{3} \arcsin \left (\frac {x}{a}\right ) - a^{2} x \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arccsc(a/x) - 1/4*(a^3*arcsin(x/a) - a^2*x*sqrt(-x^2/a^2 + 1))/a

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Fricas [A]
time = 0.34, size = 38, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - \frac {1}{4} \, {\left (a^{2} - 2 \, x^{2}\right )} \operatorname {arccsc}\left (\frac {a}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2*sqrt((a^2 - x^2)/x^2) - 1/4*(a^2 - 2*x^2)*arccsc(a/x)

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Sympy [A]
time = 0.09, size = 41, normalized size = 0.87 \begin {gather*} \begin {cases} - \frac {a^{2} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{4} + \frac {a x \sqrt {1 - \frac {x^{2}}{a^{2}}}}{4} + \frac {x^{2} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{2} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*acsc(a/x)/4 + a*x*sqrt(1 - x**2/a**2)/4 + x**2*acsc(a/x)/2, Ne(a, 0)), (zoo*x**2, True))

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Giac [A]
time = 0.41, size = 48, normalized size = 1.02 \begin {gather*} \frac {1}{2} \, a^{2} {\left (\frac {x^{2}}{a^{2}} - 1\right )} \arcsin \left (\frac {x}{a}\right ) + \frac {1}{4} \, a^{2} \arcsin \left (\frac {x}{a}\right ) + \frac {1}{4} \, a x \sqrt {-\frac {x^{2}}{a^{2}} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.61, size = 38, normalized size = 0.81 \begin {gather*} \frac {a^2\,\mathrm {asin}\left (\frac {x}{a}\right )\,\left (\frac {2\,x^2}{a^2}-1\right )}{4}+\frac {a\,x\,\sqrt {1-\frac {x^2}{a^2}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asin(x/a),x)

[Out]

(a^2*asin(x/a)*((2*x^2)/a^2 - 1))/4 + (a*x*(1 - x^2/a^2)^(1/2))/4

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