3.211 \(\int \frac{\tanh ^{-1}(\sqrt{x})}{\sqrt{x}} \, dx\)

Optimal. Leaf size=20 \[ \log (1-x)+2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*Sqrt[x]*ArcTanh[Sqrt[x]] + Log[1 - x]

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Rubi [A]  time = 0.0091329, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6097, 31} \[ \log (1-x)+2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sqrt[x]*ArcTanh[Sqrt[x]] + Log[1 - x]

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0084416, size = 20, normalized size = 1. \[ \log (1-x)+2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sqrt[x]*ArcTanh[Sqrt[x]] + Log[1 - x]

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Maple [A]  time = 0.025, size = 17, normalized size = 0.9 \begin{align*} \ln \left ( 1-x \right ) +2\,{\it Artanh} \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1-x)+2*arctanh(x^(1/2))*x^(1/2)

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Maxima [A]  time = 0.954426, size = 22, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x} \operatorname{artanh}\left (\sqrt{x}\right ) + \log \left (-x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arctanh(sqrt(x)) + log(-x + 1)

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Fricas [A]  time = 1.70806, size = 76, normalized size = 3.8 \begin{align*} \sqrt{x} \log \left (-\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right ) + \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.892376, size = 87, normalized size = 4.35 \begin{align*} \frac{2 x^{\frac{3}{2}} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} - \frac{2 \sqrt{x} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} + \frac{2 x \log{\left (\sqrt{x} + 1 \right )}}{x - 1} - \frac{2 x \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} - \frac{2 \log{\left (\sqrt{x} + 1 \right )}}{x - 1} + \frac{2 \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(3/2)*atanh(sqrt(x))/(x - 1) - 2*sqrt(x)*atanh(sqrt(x))/(x - 1) + 2*x*log(sqrt(x) + 1)/(x - 1) - 2*x*atan
h(sqrt(x))/(x - 1) - 2*log(sqrt(x) + 1)/(x - 1) + 2*atanh(sqrt(x))/(x - 1)

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Giac [A]  time = 1.18187, size = 34, normalized size = 1.7 \begin{align*} \sqrt{x} \log \left (-\frac{\sqrt{x} + 1}{\sqrt{x} - 1}\right ) + \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)*log(-(sqrt(x) + 1)/(sqrt(x) - 1)) + log(abs(x - 1))