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

Optimal. Leaf size=22 \[ \sqrt {x}-\tanh ^{-1}\left (\sqrt {x}\right )+x \coth ^{-1}\left (\sqrt {x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6092, 50, 63, 206} \[ \sqrt {x}-\tanh ^{-1}\left (\sqrt {x}\right )+x \coth ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x] + x*ArcCoth[Sqrt[x]] - ArcTanh[Sqrt[x]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6092

Int[ArcCoth[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcCoth[c*x^n], x] - Dist[c*n, Int[x^n/(1 - c^2*x^(2*n)), x]
, x] /; FreeQ[{c, n}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ \sqrt {x}-\tanh ^{-1}\left (\sqrt {x}\right )+x \coth ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x] + x*ArcCoth[Sqrt[x]] - ArcTanh[Sqrt[x]]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 24, normalized size = 1.09 \[ \frac {1}{2} \, {\left (x - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(sqrt(x)), x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 27, normalized size = 1.23 \[ x \,\mathrm {arccoth}\left (\sqrt {x}\right )+\sqrt {x}+\frac {\ln \left (-1+\sqrt {x}\right )}{2}-\frac {\ln \left (1+\sqrt {x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.30, size = 26, normalized size = 1.18 \[ x \operatorname {arcoth}\left (\sqrt {x}\right ) + \sqrt {x} - \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.24, size = 16, normalized size = 0.73 \[ x\,\mathrm {acoth}\left (\sqrt {x}\right )-\mathrm {acoth}\left (\sqrt {x}\right )+\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*acoth(x^(1/2)) - acoth(x^(1/2)) + x^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acoth}{\left (\sqrt {x} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(sqrt(x)), x)

________________________________________________________________________________________