\(\int x \csc ^{-1}(\sqrt {x}) \, dx\) [4]

Optimal result

Integrand size = 8, antiderivative size = 36 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {\sqrt {-1+x}}{2}+\frac {1}{6} (-1+x)^{3/2}+\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right ) \]

[Out]

1/6*(-1+x)^(3/2)+1/2*x^2*arccsc(x^(1/2))+1/2*(-1+x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5379, 12, 45} \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} (x-1)^{3/2}+\frac {\sqrt {x-1}}{2} \]

[In]

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

[Out]

Sqrt[-1 + x]/2 + (-1 + x)^(3/2)/6 + (x^2*ArcCsc[Sqrt[x]])/2

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}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \int \frac {x}{2 \sqrt {-1+x}} \, dx \\ & = \frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \frac {x}{\sqrt {-1+x}} \, dx \\ & = \frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \left (\frac {1}{\sqrt {-1+x}}+\sqrt {-1+x}\right ) \, dx \\ & = \frac {\sqrt {-1+x}}{2}+\frac {1}{6} (-1+x)^{3/2}+\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{6} \left (\sqrt {-1+x} (2+x)+3 x^2 \csc ^{-1}\left (\sqrt {x}\right )\right ) \]

[In]

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

[Out]

(Sqrt[-1 + x]*(2 + x) + 3*x^2*ArcCsc[Sqrt[x]])/6

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78

method	result	size
parts	\(\frac {x^{2} \operatorname {arccsc}\left (\sqrt {x}\right )}{2}+\frac {\sqrt {\frac {x -1}{x}}\, \sqrt {x}\, \left (2+x \right )}{6}\)	\(28\)
derivativedivides	\(\frac {x^{2} \operatorname {arccsc}\left (\sqrt {x}\right )}{2}+\frac {\left (x -1\right ) \left (2+x \right )}{6 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\)	\(31\)
default	\(\frac {x^{2} \operatorname {arccsc}\left (\sqrt {x}\right )}{2}+\frac {\left (x -1\right ) \left (2+x \right )}{6 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\)	\(31\)

[In]

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

[Out]

1/2*x^2*arccsc(x^(1/2))+1/6*((x-1)/x)^(1/2)*x^(1/2)*(2+x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.56 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{6} \, {\left (x + 2\right )} \sqrt {x - 1} \]

[In]

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

[Out]

1/2*x^2*arccsc(sqrt(x)) + 1/6*(x + 2)*sqrt(x - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.87 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x^{2} \operatorname {acsc}{\left (\sqrt {x} \right )}}{2} + \frac {\begin {cases} \frac {2 x \sqrt {x - 1}}{3} + \frac {4 \sqrt {x - 1}}{3} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x \sqrt {1 - x}}{3} + \frac {4 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}}{4} \]

[In]

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

[Out]

x**2*acsc(sqrt(x))/2 + Piecewise((2*x*sqrt(x - 1)/3 + 4*sqrt(x - 1)/3, Abs(x) > 1), (2*I*x*sqrt(1 - x)/3 + 4*I
*sqrt(1 - x)/3, True))/4

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{6} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{2} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]

[In]

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

[Out]

1/6*x^(3/2)*(-1/x + 1)^(3/2) + 1/2*x^2*arccsc(sqrt(x)) + 1/2*sqrt(x)*sqrt(-1/x + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).

Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.22 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{48} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} + \frac {1}{2} \, x^{2} \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {3}{16} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {9 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 1}{48 \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3}} \]

[In]

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

[Out]

1/48*x^(3/2)*(sqrt(-1/x + 1) - 1)^3 + 1/2*x^2*arcsin(1/sqrt(x)) + 3/16*sqrt(x)*(sqrt(-1/x + 1) - 1) - 1/48*(9*
x*(sqrt(-1/x + 1) - 1)^2 + 1)/(x^(3/2)*(sqrt(-1/x + 1) - 1)^3)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x*asin(1/x^(1/2)), x)