3.3.11 \(\int \frac {\tanh ^{-1}(\sqrt {x})}{\sqrt {x}} \, dx\) [211]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {6037, 31} \begin {gather*} \log (1-x)+2 \sqrt {x} \tanh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

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 )}{\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 = 0.01, size = 20, normalized size = 1.00 \begin {gather*} 2 \sqrt {x} \tanh ^{-1}\left (\sqrt {x}\right )+\log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 17, normalized size = 0.85

method	result	size
derivativedivides	\(\ln \left (1-x \right )+2 \arctanh \left (\sqrt {x}\right ) \sqrt {x}\)	\(17\)
default	\(\ln \left (1-x \right )+2 \arctanh \left (\sqrt {x}\right ) \sqrt {x}\)	\(17\)
meijerg	\(-\sqrt {x}\, \left (\ln \left (1-\sqrt {x}\right )-\ln \left (\sqrt {x}+1\right )\right )+\ln \left (1-x \right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 16, normalized size = 0.80 \begin {gather*} 2 \, \sqrt {x} \operatorname {artanh}\left (\sqrt {x}\right ) + \log \left (-x + 1\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 25, normalized size = 1.25 \begin {gather*} \sqrt {x} \log \left (-\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \log \left (x - 1\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (17) = 34\).
time = 0.21, size = 87, normalized size = 4.35 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (16) = 32\).
time = 0.47, size = 72, normalized size = 3.60 \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^(1/2),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))

________________________________________________________________________________________

Mupad [B]
time = 0.80, size = 14, normalized size = 0.70 \begin {gather*} \ln \left (x-1\right )+2\,\sqrt {x}\,\mathrm {atanh}\left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 1) + 2*x^(1/2)*atanh(x^(1/2))

________________________________________________________________________________________