\(\int x^2 \csc ^{-1}(\frac {a}{x}) \, dx\) [9]

Optimal result

Integrand size = 10, antiderivative size = 56 \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} x^3 \arcsin \left (\frac {x}{a}\right ) \]

[Out]

-1/9*a^3*(1-x^2/a^2)^(3/2)+1/3*x^3*arcsin(x/a)+1/3*a^3*(1-x^2/a^2)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5373, 4723, 272, 45} \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=-\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{3} x^3 \arcsin \left (\frac {x}{a}\right ) \]

[In]

Int[x^2*ArcCsc[a/x],x]

[Out]

(a^3*Sqrt[1 - x^2/a^2])/3 - (a^3*(1 - x^2/a^2)^(3/2))/9 + (x^3*ArcSin[x/a])/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

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*} \text {integral}& = \int x^2 \arcsin \left (\frac {x}{a}\right ) \, dx \\ & = \frac {1}{3} x^3 \arcsin \left (\frac {x}{a}\right )-\frac {\int \frac {x^3}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{3 a} \\ & = \frac {1}{3} x^3 \arcsin \left (\frac {x}{a}\right )-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x}{a^2}}} \, dx,x,x^2\right )}{6 a} \\ & = \frac {1}{3} x^3 \arcsin \left (\frac {x}{a}\right )-\frac {\text {Subst}\left (\int \left (\frac {a^2}{\sqrt {1-\frac {x}{a^2}}}-a^2 \sqrt {1-\frac {x}{a^2}}\right ) \, dx,x,x^2\right )}{6 a} \\ & = \frac {1}{3} a^3 \sqrt {1-\frac {x^2}{a^2}}-\frac {1}{9} a^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}+\frac {1}{3} x^3 \arcsin \left (\frac {x}{a}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{9} a \left (2 a^2+x^2\right ) \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{3} x^3 \csc ^{-1}\left (\frac {a}{x}\right ) \]

[In]

Integrate[x^2*ArcCsc[a/x],x]

[Out]

(a*(2*a^2 + x^2)*Sqrt[1 - x^2/a^2])/9 + (x^3*ArcCsc[a/x])/3

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00

method	result	size
parts	\(\frac {x^{3} \operatorname {arccsc}\left (\frac {a}{x}\right )}{3}-\frac {-\frac {x^{2} a^{2} \sqrt {1-\frac {x^{2}}{a^{2}}}}{3}-\frac {2 a^{4} \sqrt {1-\frac {x^{2}}{a^{2}}}}{3}}{3 a}\)	\(56\)
derivativedivides	\(-a^{3} \left (-\frac {x^{3} \operatorname {arccsc}\left (\frac {a}{x}\right )}{3 a^{3}}-\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) \left (\frac {2 a^{2}}{x^{2}}+1\right ) x^{4}}{9 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{4}}\right )\)	\(66\)
default	\(-a^{3} \left (-\frac {x^{3} \operatorname {arccsc}\left (\frac {a}{x}\right )}{3 a^{3}}-\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) \left (\frac {2 a^{2}}{x^{2}}+1\right ) x^{4}}{9 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{4}}\right )\)	\(66\)

[In]

int(x^2*arccsc(a/x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.70 \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arccsc}\left (\frac {a}{x}\right ) + \frac {1}{9} \, {\left (2 \, a^{2} x + x^{3}\right )} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} \]

[In]

integrate(x^2*arccsc(a/x),x, algorithm="fricas")

[Out]

1/3*x^3*arccsc(a/x) + 1/9*(2*a^2*x + x^3)*sqrt((a^2 - x^2)/x^2)

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=\begin {cases} \frac {2 a^{3} \sqrt {1 - \frac {x^{2}}{a^{2}}}}{9} + \frac {a x^{2} \sqrt {1 - \frac {x^{2}}{a^{2}}}}{9} + \frac {x^{3} \operatorname {acsc}{\left (\frac {a}{x} \right )}}{3} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{3} & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*acsc(a/x),x)

[Out]

Piecewise((2*a**3*sqrt(1 - x**2/a**2)/9 + a*x**2*sqrt(1 - x**2/a**2)/9 + x**3*acsc(a/x)/3, Ne(a, 0)), (zoo*x**
3, True))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arccsc}\left (\frac {a}{x}\right ) + \frac {2 \, a^{4} \sqrt {-\frac {x^{2}}{a^{2}} + 1} + a^{2} x^{2} \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{9 \, a} \]

[In]

integrate(x^2*arccsc(a/x),x, algorithm="maxima")

[Out]

1/3*x^3*arccsc(a/x) + 1/9*(2*a^4*sqrt(-x^2/a^2 + 1) + a^2*x^2*sqrt(-x^2/a^2 + 1))/a

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{3} \, a^{2} x {\left (\frac {x^{2}}{a^{2}} - 1\right )} \arcsin \left (\frac {x}{a}\right ) - \frac {1}{9} \, a^{3} {\left (-\frac {x^{2}}{a^{2}} + 1\right )}^{\frac {3}{2}} + \frac {1}{3} \, a^{2} x \arcsin \left (\frac {x}{a}\right ) + \frac {1}{3} \, a^{3} \sqrt {-\frac {x^{2}}{a^{2}} + 1} \]

[In]

integrate(x^2*arccsc(a/x),x, algorithm="giac")

[Out]

1/3*a^2*x*(x^2/a^2 - 1)*arcsin(x/a) - 1/9*a^3*(-x^2/a^2 + 1)^(3/2) + 1/3*a^2*x*arcsin(x/a) + 1/3*a^3*sqrt(-x^2
/a^2 + 1)

Mupad [F(-1)]

Timed out. \[ \int x^2 \csc ^{-1}\left (\frac {a}{x}\right ) \, dx=\left \{\begin {array}{cl} \frac {x^3\,\mathrm {asin}\left (\frac {x}{a}\right )}{3}+\frac {\sqrt {a^2-x^2}\,\left (2\,a^2+x^2\right )}{9} & \text {\ if\ \ }0<a\\ \int x^2\,\mathrm {asin}\left (\frac {x}{a}\right ) \,d x & \text {\ if\ \ }\neg 0<a \end {array}\right . \]

[In]

int(x^2*asin(x/a),x)

[Out]

piecewise(0 < a, (x^3*asin(x/a))/3 + ((a^2 - x^2)^(1/2)*(2*a^2 + x^2))/9, ~0 < a, int(x^2*asin(x/a), x))