3.5 \(\int \csc ^{-1}(\sqrt {x}) \, dx\)

Optimal. Leaf size=16 \[ \sqrt {x-1}+x \csc ^{-1}\left (\sqrt {x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5269, 12, 32} \[ \sqrt {x-1}+x \csc ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[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 5269

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

Rubi steps

\begin {align*} \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx &=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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \sqrt {x-1}+x \csc ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x] + x*ArcCsc[Sqrt[x]]

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fricas [A]  time = 1.86, size = 12, normalized size = 0.75 \[ x \operatorname {arccsc}\left (\sqrt {x}\right ) + \sqrt {x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccsc(sqrt(x)) + sqrt(x - 1)

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giac [B]  time = 0.15, size = 41, normalized size = 2.56 \[ x \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {1}{2} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {1}{2 \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.05, size = 24, normalized size = 1.50 \[ x \,\mathrm {arccsc}\left (\sqrt {x}\right )+\frac {x -1}{\sqrt {\frac {x -1}{x}}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 20, normalized size = 1.25 \[ x \operatorname {arccsc}\left (\sqrt {x}\right ) + \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.06, size = 20, normalized size = 1.25 \[ x\,\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right )+\sqrt {x}\,\sqrt {1-\frac {1}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 3.71, size = 29, normalized size = 1.81 \[ x \operatorname {acsc}{\left (\sqrt {x} \right )} + \frac {\begin {cases} 2 \sqrt {x - 1} & \text {for}\: \left |{x}\right | > 1 \\2 i \sqrt {1 - x} & \text {otherwise} \end {cases}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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