3.90 \(\int \sqrt{x} \coth ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{3} x^{3/2} \coth ^{-1}\left (\sqrt{x}\right )+\frac{x}{3}+\frac{1}{3} \log (1-x) \]

[Out]

x/3 + (2*x^(3/2)*ArcCoth[Sqrt[x]])/3 + Log[1 - x]/3

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Rubi [A]  time = 0.0135038, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6098, 43} \[ \frac{2}{3} x^{3/2} \coth ^{-1}\left (\sqrt{x}\right )+\frac{x}{3}+\frac{1}{3} \log (1-x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*ArcCoth[Sqrt[x]],x]

[Out]

x/3 + (2*x^(3/2)*ArcCoth[Sqrt[x]])/3 + Log[1 - x]/3

Rule 6098

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
th[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

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

Mathematica [A]  time = 0.0101959, size = 25, normalized size = 0.81 \[ \frac{1}{3} \left (2 x^{3/2} \coth ^{-1}\left (\sqrt{x}\right )+x+\log (1-x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*ArcCoth[Sqrt[x]],x]

[Out]

(x + 2*x^(3/2)*ArcCoth[Sqrt[x]] + Log[1 - x])/3

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Maple [A]  time = 0.036, size = 30, normalized size = 1. \begin{align*}{\frac{2}{3}{x}^{{\frac{3}{2}}}{\rm arccoth} \left (\sqrt{x}\right )}+{\frac{x}{3}}+{\frac{1}{3}\ln \left ( -1+\sqrt{x} \right ) }+{\frac{1}{3}\ln \left ( 1+\sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arccoth(x^(1/2))+1/3*x+1/3*ln(-1+x^(1/2))+1/3*ln(1+x^(1/2))

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Maxima [A]  time = 0.990299, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \operatorname{arcoth}\left (\sqrt{x}\right ) + \frac{1}{3} \, x + \frac{1}{3} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arccoth(sqrt(x)) + 1/3*x + 1/3*log(x - 1)

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Fricas [A]  time = 1.65532, size = 96, normalized size = 3.1 \begin{align*} \frac{1}{3} \, x^{\frac{3}{2}} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right ) + \frac{1}{3} \, x + \frac{1}{3} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.60655, size = 39, normalized size = 1.26 \begin{align*} \frac{2 x^{\frac{3}{2}} \operatorname{acoth}{\left (\sqrt{x} \right )}}{3} + \frac{x}{3} + \frac{2 \log{\left (\sqrt{x} + 1 \right )}}{3} - \frac{2 \operatorname{acoth}{\left (\sqrt{x} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(3/2)*acoth(sqrt(x))/3 + x/3 + 2*log(sqrt(x) + 1)/3 - 2*acoth(sqrt(x))/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x)*arccoth(sqrt(x)), x)