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

Optimal. Leaf size=36 \[ \frac{1}{2} x^2 \csc ^{-1}\left (\sqrt{x}\right )+\frac{1}{6} (x-1)^{3/2}+\frac{\sqrt{x-1}}{2} \]

[Out]

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

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Rubi [A]  time = 0.0119593, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5271, 12, 43} \[ \frac{1}{2} x^2 \csc ^{-1}\left (\sqrt{x}\right )+\frac{1}{6} (x-1)^{3/2}+\frac{\sqrt{x-1}}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5271

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*
Sqrt[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]

Rule 12

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

Rule 43

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])

Rubi steps

\begin{align*} \int x \csc ^{-1}\left (\sqrt{x}\right ) \, dx &=\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]  time = 0.0207999, size = 27, normalized size = 0.75 \[ \frac{1}{6} \left (3 x^2 \csc ^{-1}\left (\sqrt{x}\right )+\sqrt{x-1} (x+2)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.114, size = 31, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}{\rm arccsc} \left (\sqrt{x}\right )}+{\frac{ \left ( x-1 \right ) \left ( x+2 \right ) }{6}{\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(x*arccsc(x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.01095, size = 51, normalized size = 1.42 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 3.20773, size = 69, normalized size = 1.92 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arccsc}\left (\sqrt{x}\right ) + \frac{1}{6} \,{\left (x + 2\right )} \sqrt{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*acsc(sqrt(x)), x)

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Giac [A]  time = 1.11054, size = 36, normalized size = 1. \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (\frac{1}{\sqrt{x}}\right ) + \frac{1}{6} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{1}{3} \, i + \frac{1}{2} \, \sqrt{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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