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

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

[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 = 0.02, 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 = {5265, 4627, 321, 216} \[ \frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{4} a^2 \sin ^{-1}\left (\frac {x}{a}\right )+\frac {1}{2} x^2 \sin ^{-1}\left (\frac {x}{a}\right ) \]

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 216

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 321

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 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 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 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 \[ \frac {1}{4} \left (a x \sqrt {1-\frac {x^2}{a^2}}+a^2 \left (-\sin ^{-1}\left (\frac {x}{a}\right )\right )+2 x^2 \csc ^{-1}\left (\frac {a}{x}\right )\right ) \]

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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fricas [A]  time = 0.80, size = 38, normalized size = 0.81 \[ \frac {1}{4} \, x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - \frac {1}{4} \, {\left (a^{2} - 2 \, x^{2}\right )} \operatorname {arccsc}\left (\frac {a}{x}\right ) \]

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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giac [A]  time = 0.14, size = 48, normalized size = 1.02 \[ \frac {1}{2} \, a^{2} {\left (\frac {x^{2}}{a^{2}} - 1\right )} \arcsin \left (\frac {x}{a}\right ) + \frac {1}{4} \, a^{2} \arcsin \left (\frac {x}{a}\right ) + \frac {1}{4} \, a x \sqrt {-\frac {x^{2}}{a^{2}} + 1} \]

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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maple [B]  time = 0.06, size = 93, normalized size = 1.98 \[ \frac {x^{2} \mathrm {arccsc}\left (\frac {a}{x}\right )}{2}-\frac {a \sqrt {-1+\frac {a^{2}}{x^{2}}}\, x \arctan \left (\frac {1}{\sqrt {-1+\frac {a^{2}}{x^{2}}}}\right )}{4 \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}}+\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{3}}{4 a \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 46, normalized size = 0.98 \[ \frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\frac {a}{x}\right ) - \frac {a^{3} \arcsin \left (\frac {x}{a}\right ) - a^{2} x \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{4 \, a} \]

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

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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sympy [A]  time = 0.21, size = 41, normalized size = 0.87 \[ \begin {cases} - \frac {a^{2} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{4} + \frac {a x \sqrt {1 - \frac {x^{2}}{a^{2}}}}{4} + \frac {x^{2} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{2} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{2} & \text {otherwise} \end {cases} \]

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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