3.1.90 \(\int \sqrt {x} \coth ^{-1}(\sqrt {x}) \, dx\) [90]

Optimal. Leaf size=31 \[ \frac {x}{3}+\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \log (1-x) \]

[Out]

1/3*x+2/3*x^(3/2)*arccoth(x^(1/2))+1/3*ln(1-x)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6038, 45} \begin {gather*} \frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {x}{3}+\frac {1}{3} \log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/3 + (2*x^(3/2)*ArcCoth[Sqrt[x]])/3 + Log[1 - x]/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6038

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCoth[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCoth[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 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x}{1-x} \, dx\\ &=\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \left (-1+\frac {1}{1-x}\right ) \, dx\\ &=\frac {x}{3}+\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \log (1-x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 25, normalized size = 0.81 \begin {gather*} \frac {1}{3} \left (x+2 x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\log (1-x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x + 2*x^(3/2)*ArcCoth[Sqrt[x]] + Log[1 - x])/3

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 30, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}} \mathrm {arccoth}\left (\sqrt {x}\right )}{3}+\frac {x}{3}+\frac {\ln \left (\sqrt {x}-1\right )}{3}+\frac {\ln \left (\sqrt {x}+1\right )}{3}\)	\(30\)
default	\(\frac {2 x^{\frac {3}{2}} \mathrm {arccoth}\left (\sqrt {x}\right )}{3}+\frac {x}{3}+\frac {\ln \left (\sqrt {x}-1\right )}{3}+\frac {\ln \left (\sqrt {x}+1\right )}{3}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arccoth(x^(1/2))+1/3*x+1/3*ln(x^(1/2)-1)+1/3*ln(x^(1/2)+1)

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 19, normalized size = 0.61 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{3} \, x^{\frac {3}{2}} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.56, size = 39, normalized size = 1.26 \begin {gather*} \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{3} + \frac {x}{3} + \frac {2 \log {\left (\sqrt {x} + 1 \right )}}{3} - \frac {2 \operatorname {acoth}{\left (\sqrt {x} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (21) = 42\).
time = 0.39, size = 119, normalized size = 3.84 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + 1\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{3 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{3}} + \frac {4 \, {\left (\sqrt {x} + 1\right )}}{3 \, {\left (\sqrt {x} - 1\right )} {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{2}} + \frac {2}{3} \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) - \frac {2}{3} \, \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(arccoth(x^(1/2))*x^(1/2),x, algorithm="giac")

[Out]

2/3*(3*(sqrt(x) + 1)^2/(sqrt(x) - 1)^2 + 1)*log((sqrt(x) + 1)/(sqrt(x) - 1))/((sqrt(x) + 1)/(sqrt(x) - 1) - 1)
^3 + 4/3*(sqrt(x) + 1)/((sqrt(x) - 1)*((sqrt(x) + 1)/(sqrt(x) - 1) - 1)^2) + 2/3*log((sqrt(x) + 1)/abs(sqrt(x)
- 1)) - 2/3*log(abs((sqrt(x) + 1)/(sqrt(x) - 1) - 1))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {x}\,\mathrm {acoth}\left (\sqrt {x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^(1/2)*acoth(x^(1/2)), x)

________________________________________________________________________________________