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

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

[Out]

Sqrt[-1 + x] + x*ArcCsc[Sqrt[x]]

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Rubi [A]  time = 0.0039246, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 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]

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]

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*}

Mathematica [A]  time = 0.0043057, size = 16, normalized size = 1. \[ \sqrt{x-1}+x \csc ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x] + x*ArcCsc[Sqrt[x]]

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Maple [A]  time = 0.116, size = 24, normalized size = 1.5 \begin{align*} x{\rm arccsc} \left (\sqrt{x}\right )+{(x-1){\frac{1}{\sqrt{{\frac{x-1}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}

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 = 1.01964, size = 27, normalized size = 1.69 \begin{align*} x \operatorname{arccsc}\left (\sqrt{x}\right ) + \sqrt{x} \sqrt{-\frac{1}{x} + 1} \end{align*}

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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Fricas [A]  time = 3.08308, size = 45, normalized size = 2.81 \begin{align*} x \operatorname{arccsc}\left (\sqrt{x}\right ) + \sqrt{x - 1} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acsc}{\left (\sqrt{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acsc(sqrt(x)), x)

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Giac [A]  time = 1.1163, size = 20, normalized size = 1.25 \begin{align*} x \arcsin \left (\frac{1}{\sqrt{x}}\right ) - i + \sqrt{x - 1} \end{align*}

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)) - i + sqrt(x - 1)