3.212 \(\int \frac{\tanh ^{-1}(\sqrt{x})}{x^{3/2}} \, dx\)

Optimal. Leaf size=24 \[ -\log (1-x)+\log (x)-\frac{2 \tanh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

[Out]

(-2*ArcTanh[Sqrt[x]])/Sqrt[x] - Log[1 - x] + Log[x]

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Rubi [A]  time = 0.0102569, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6097, 36, 31, 29} \[ -\log (1-x)+\log (x)-\frac{2 \tanh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Sqrt[x]]/x^(3/2),x]

[Out]

(-2*ArcTanh[Sqrt[x]])/Sqrt[x] - Log[1 - x] + Log[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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0176779, size = 24, normalized size = 1. \[ -\log (1-x)+\log (x)-\frac{2 \tanh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Sqrt[x]]/x^(3/2),x]

[Out]

(-2*ArcTanh[Sqrt[x]])/Sqrt[x] - Log[1 - x] + Log[x]

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Maple [A]  time = 0.03, size = 29, normalized size = 1.2 \begin{align*} -2\,{\frac{{\it Artanh} \left ( \sqrt{x} \right ) }{\sqrt{x}}}-\ln \left ( -1+\sqrt{x} \right ) +\ln \left ( x \right ) -\ln \left ( 1+\sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x^(1/2))/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.958226, size = 24, normalized size = 1. \begin{align*} -\frac{2 \, \operatorname{artanh}\left (\sqrt{x}\right )}{\sqrt{x}} - \log \left (x - 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x^(1/2))/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.80816, size = 100, normalized size = 4.17 \begin{align*} -\frac{x \log \left (x - 1\right ) - x \log \left (x\right ) + \sqrt{x} \log \left (-\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x^(1/2))/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.41666, size = 126, normalized size = 5.25 \begin{align*} - \frac{2 x^{\frac{3}{2}} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x^{2} - x} + \frac{2 \sqrt{x} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x^{2} - x} + \frac{x^{2} \log{\left (x \right )}}{x^{2} - x} - \frac{2 x^{2} \log{\left (\sqrt{x} + 1 \right )}}{x^{2} - x} + \frac{2 x^{2} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x^{2} - x} - \frac{x \log{\left (x \right )}}{x^{2} - x} + \frac{2 x \log{\left (\sqrt{x} + 1 \right )}}{x^{2} - x} - \frac{2 x \operatorname{atanh}{\left (\sqrt{x} \right )}}{x^{2} - x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(x**(1/2))/x**(3/2),x)

[Out]

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

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Giac [A]  time = 1.16425, size = 41, normalized size = 1.71 \begin{align*} -\frac{\log \left (-\frac{\sqrt{x} + 1}{\sqrt{x} - 1}\right )}{\sqrt{x}} + \log \left (x\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^(3/2),x, algorithm="giac")

[Out]

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