\(\int x^2 \csc ^{-1}(\sqrt {x}) \, dx\) [3]

Optimal result

Integrand size = 10, antiderivative size = 47 \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {\sqrt {-1+x}}{3}+\frac {2}{9} (-1+x)^{3/2}+\frac {1}{15} (-1+x)^{5/2}+\frac {1}{3} x^3 \csc ^{-1}\left (\sqrt {x}\right ) \]

[Out]

2/9*(-1+x)^(3/2)+1/15*(-1+x)^(5/2)+1/3*x^3*arccsc(x^(1/2))+1/3*(-1+x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5379, 12, 45} \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} x^3 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{15} (x-1)^{5/2}+\frac {2}{9} (x-1)^{3/2}+\frac {\sqrt {x-1}}{3} \]

[In]

Int[x^2*ArcCsc[Sqrt[x]],x]

[Out]

Sqrt[-1 + x]/3 + (2*(-1 + x)^(3/2))/9 + (-1 + x)^(5/2)/15 + (x^3*ArcCsc[Sqrt[x]])/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 5379

Int[((a_.) + ArcCsc[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcCsc[
u])/(d*(m + 1))), x] + Dist[b*(u/(d*(m + 1)*Sqrt[u^2])), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(u*S
qrt[u^2 - 1])), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !Funct
ionOfQ[(c + d*x)^(m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \int \frac {x^2}{2 \sqrt {-1+x}} \, dx \\ & = \frac {1}{3} x^3 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} \int \frac {x^2}{\sqrt {-1+x}} \, dx \\ & = \frac {1}{3} x^3 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} \int \left (\frac {1}{\sqrt {-1+x}}+2 \sqrt {-1+x}+(-1+x)^{3/2}\right ) \, dx \\ & = \frac {\sqrt {-1+x}}{3}+\frac {2}{9} (-1+x)^{3/2}+\frac {1}{15} (-1+x)^{5/2}+\frac {1}{3} x^3 \csc ^{-1}\left (\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{45} \sqrt {-1+x} \left (8+4 x+3 x^2\right )+\frac {1}{3} x^3 \csc ^{-1}\left (\sqrt {x}\right ) \]

[In]

Integrate[x^2*ArcCsc[Sqrt[x]],x]

[Out]

(Sqrt[-1 + x]*(8 + 4*x + 3*x^2))/45 + (x^3*ArcCsc[Sqrt[x]])/3

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74

method	result	size
parts	\(\frac {x^{3} \operatorname {arccsc}\left (\sqrt {x}\right )}{3}+\frac {\sqrt {\frac {x -1}{x}}\, \sqrt {x}\, \left (3 x^{2}+4 x +8\right )}{45}\)	\(35\)
derivativedivides	\(\frac {x^{3} \operatorname {arccsc}\left (\sqrt {x}\right )}{3}+\frac {\left (x -1\right ) \left (3 x^{2}+4 x +8\right )}{45 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\)	\(38\)
default	\(\frac {x^{3} \operatorname {arccsc}\left (\sqrt {x}\right )}{3}+\frac {\left (x -1\right ) \left (3 x^{2}+4 x +8\right )}{45 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\)	\(38\)

[In]

int(x^2*arccsc(x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

1/3*x^3*arccsc(x^(1/2))+1/45*((x-1)/x)^(1/2)*x^(1/2)*(3*x^2+4*x+8)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.57 \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{45} \, {\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt {x - 1} \]

[In]

integrate(x^2*arccsc(x^(1/2)),x, algorithm="fricas")

[Out]

1/3*x^3*arccsc(sqrt(x)) + 1/45*(3*x^2 + 4*x + 8)*sqrt(x - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 19.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.91 \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x^{3} \operatorname {acsc}{\left (\sqrt {x} \right )}}{3} + \frac {\begin {cases} \frac {2 x^{2} \sqrt {x - 1}}{5} + \frac {8 x \sqrt {x - 1}}{15} + \frac {16 \sqrt {x - 1}}{15} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x^{2} \sqrt {1 - x}}{5} + \frac {8 i x \sqrt {1 - x}}{15} + \frac {16 i \sqrt {1 - x}}{15} & \text {otherwise} \end {cases}}{6} \]

[In]

integrate(x**2*acsc(x**(1/2)),x)

[Out]

x**3*acsc(sqrt(x))/3 + Piecewise((2*x**2*sqrt(x - 1)/5 + 8*x*sqrt(x - 1)/15 + 16*sqrt(x - 1)/15, Abs(x) > 1),
(2*I*x**2*sqrt(1 - x)/5 + 8*I*x*sqrt(1 - x)/15 + 16*I*sqrt(1 - x)/15, True))/6

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.11 \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{15} \, x^{\frac {5}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{3} \, x^{3} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {2}{9} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{3} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]

[In]

integrate(x^2*arccsc(x^(1/2)),x, algorithm="maxima")

[Out]

1/15*x^(5/2)*(-1/x + 1)^(5/2) + 1/3*x^3*arccsc(sqrt(x)) + 2/9*x^(3/2)*(-1/x + 1)^(3/2) + 1/3*sqrt(x)*sqrt(-1/x
+ 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.47 \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{480} \, x^{\frac {5}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{5} + \frac {5}{288} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} + \frac {1}{3} \, x^{3} \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {5}{48} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {150 \, x^{2} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{4} + 25 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 3}{1440 \, x^{\frac {5}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{5}} \]

[In]

integrate(x^2*arccsc(x^(1/2)),x, algorithm="giac")

[Out]

1/480*x^(5/2)*(sqrt(-1/x + 1) - 1)^5 + 5/288*x^(3/2)*(sqrt(-1/x + 1) - 1)^3 + 1/3*x^3*arcsin(1/sqrt(x)) + 5/48
*sqrt(x)*(sqrt(-1/x + 1) - 1) - 1/1440*(150*x^2*(sqrt(-1/x + 1) - 1)^4 + 25*x*(sqrt(-1/x + 1) - 1)^2 + 3)/(x^(
5/2)*(sqrt(-1/x + 1) - 1)^5)

Mupad [F(-1)]

Timed out. \[ \int x^2 \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\int x^2\,\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]

[In]

int(x^2*asin(1/x^(1/2)),x)

[Out]

int(x^2*asin(1/x^(1/2)), x)