\(\int \csc ^{-1}(\sqrt {x}) \, dx\) [5]

Optimal result

Integrand size = 6, antiderivative size = 16 \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\sqrt {-1+x}+x \csc ^{-1}\left (\sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5377, 12, 32} \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\sqrt {x-1}+x \csc ^{-1}\left (\sqrt {x}\right ) \]

[In]

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

[Out]

Sqrt[-1 + x] + x*ArcCsc[Sqrt[x]]

Rule 12

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 5377

Int[ArcCsc[u_], x_Symbol] :> Simp[x*ArcCsc[u], x] + Dist[u/Sqrt[u^2], Int[SimplifyIntegrand[x*(D[u, x]/(u*Sqrt
[u^2 - 1])), x], x], x] /; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\sqrt {-1+x}+x \csc ^{-1}\left (\sqrt {x}\right ) \]

[In]

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

[Out]

Sqrt[-1 + x] + x*ArcCsc[Sqrt[x]]

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31

method	result	size
parts	\(x \,\operatorname {arccsc}\left (\sqrt {x}\right )+\sqrt {\frac {x -1}{x}}\, \sqrt {x}\)	\(21\)
derivativedivides	\(x \,\operatorname {arccsc}\left (\sqrt {x}\right )+\frac {x -1}{\sqrt {\frac {x -1}{x}}\, \sqrt {x}}\)	\(24\)
default	\(x \,\operatorname {arccsc}\left (\sqrt {x}\right )+\frac {x -1}{\sqrt {\frac {x -1}{x}}\, \sqrt {x}}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=x \operatorname {arccsc}\left (\sqrt {x}\right ) + \sqrt {x - 1} \]

[In]

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

[Out]

x*arccsc(sqrt(x)) + sqrt(x - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=x \operatorname {acsc}{\left (\sqrt {x} \right )} + \frac {\begin {cases} 2 \sqrt {x - 1} & \text {for}\: \left |{x}\right | > 1 \\2 i \sqrt {1 - x} & \text {otherwise} \end {cases}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=x \operatorname {arccsc}\left (\sqrt {x}\right ) + \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]

[In]

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

[Out]

x*arccsc(sqrt(x)) + sqrt(x)*sqrt(-1/x + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.56 \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=x \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {1}{2} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {1}{2 \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}} \]

[In]

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

[Out]

x*arcsin(1/sqrt(x)) + 1/2*sqrt(x)*(sqrt(-1/x + 1) - 1) - 1/2/(sqrt(x)*(sqrt(-1/x + 1) - 1))

Mupad [B] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx=x\,\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right )+\sqrt {x}\,\sqrt {1-\frac {1}{x}} \]

[In]

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

[Out]

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