3.3.12 \(\int \frac {\tanh ^{-1}(\sqrt {x})}{x^{3/2}} \, dx\) [212]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {6037, 36, 31, 29} \begin {gather*} -\log (1-x)+\log (x)-\frac {2 \tanh ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \end {gather*}

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 29

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

Rule 31

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

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 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

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*}

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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\log (1-x)+\log (x) \end {gather*}

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 = 0.07, size = 29, normalized size = 1.21

method	result	size
derivativedivides	\(-\frac {2 \arctanh \left (\sqrt {x}\right )}{\sqrt {x}}+\ln \left (x \right )-\ln \left (\sqrt {x}+1\right )-\ln \left (\sqrt {x}-1\right )\)	\(29\)
default	\(-\frac {2 \arctanh \left (\sqrt {x}\right )}{\sqrt {x}}+\ln \left (x \right )-\ln \left (\sqrt {x}+1\right )-\ln \left (\sqrt {x}-1\right )\)	\(29\)
meijerg	\(\frac {\ln \left (1-\sqrt {x}\right )-\ln \left (\sqrt {x}+1\right )}{\sqrt {x}}-\ln \left (1-x \right )+\ln \left (x \right )+i \pi \)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x^(1/2))/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 18, normalized size = 0.75 \begin {gather*} -\frac {2 \, \operatorname {artanh}\left (\sqrt {x}\right )}{\sqrt {x}} - \log \left (x - 1\right ) + \log \left (x\right ) \end {gather*}

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 = 0.38, size = 37, normalized size = 1.54 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (20) = 40\).
time = 0.42, size = 126, normalized size = 5.25 \begin {gather*} - \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (20) = 40\).
time = 0.42, size = 72, normalized size = 3.00 \begin {gather*} \frac {2 \, \log \left (-\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1} - 2 \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) + 2 \, \log \left ({\left | -\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.79, size = 22, normalized size = 0.92 \begin {gather*} 2\,\ln \left (\sqrt {x}\right )-\ln \left (x-1\right )-\frac {2\,\mathrm {atanh}\left (\sqrt {x}\right )}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x^(1/2)) - log(x - 1) - (2*atanh(x^(1/2)))/x^(1/2)

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