3.1.94 \(\int \coth ^{-1}(\frac {1}{x}) \, dx\) [94]

Optimal. Leaf size=19 \[ x \coth ^{-1}\left (\frac {1}{x}\right )+\frac {1}{2} \log \left (1-x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6022, 269, 266} \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )+x \coth ^{-1}\left (\frac {1}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[x^(-1)],x]

[Out]

x*ArcCoth[x^(-1)] + Log[1 - x^2]/2

Rule 266

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

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 6022

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} x \coth ^{-1}\left (\frac {1}{x}\right )+\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[x^(-1)],x]

[Out]

x*ArcCoth[x^(-1)] + Log[1 - x^2]/2

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Maple [A]
time = 0.14, size = 30, normalized size = 1.58

method	result	size
derivativedivides	\(x \,\mathrm {arccoth}\left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}-1\right )}{2}-\ln \left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}+1\right )}{2}\)	\(30\)
default	\(x \,\mathrm {arccoth}\left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}-1\right )}{2}-\ln \left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}+1\right )}{2}\)	\(30\)
risch	\(\frac {x \ln \left (1+x \right )}{2}-\frac {\ln \left (-1+x \right ) x}{2}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (-1+x \right )}{x}\right )^{2} x}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left (1+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (1+x \right )}{x}\right )^{2} x}{4}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-1+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-1+x \right )}{x}\right ) x}{4}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (1+x \right )}{x}\right )^{3} x}{4}-\frac {i \pi x}{2}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (1+x \right )}{x}\right )^{2} x}{4}-\frac {i \pi \,\mathrm {csgn}\left (i \left (-1+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-1+x \right )}{x}\right )^{2} x}{4}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (-1+x \right )}{x}\right )^{3} x}{4}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-1+x \right )}{x}\right )^{2} x}{4}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (1+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (1+x \right )}{x}\right ) x}{4}+\frac {\ln \left (x^{2}-1\right )}{2}\)	\(234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(1/x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 15, normalized size = 0.79 \begin {gather*} x \operatorname {arcoth}\left (\frac {1}{x}\right ) + \frac {1}{2} \, \log \left (x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccoth(1/x) + 1/2*log(x^2 - 1)

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Fricas [A]
time = 0.39, size = 23, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, x \log \left (-\frac {x + 1}{x - 1}\right ) + \frac {1}{2} \, \log \left (x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 15, normalized size = 0.79 \begin {gather*} x \operatorname {acoth}{\left (\frac {1}{x} \right )} + \log {\left (x + 1 \right )} - \operatorname {acoth}{\left (\frac {1}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(1/x),x)

[Out]

x*acoth(1/x) + log(x + 1) - acoth(1/x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (17) = 34\).
time = 0.39, size = 104, normalized size = 5.47 \begin {gather*} \frac {\log \left (-\frac {\frac {\frac {x + 1}{x - 1} + 1}{\frac {x + 1}{x - 1} - 1} + 1}{\frac {\frac {x + 1}{x - 1} + 1}{\frac {x + 1}{x - 1} - 1} - 1}\right )}{\frac {x + 1}{x - 1} - 1} + \log \left (\frac {{\left | -x - 1 \right |}}{{\left | x - 1 \right |}}\right ) - \log \left ({\left | -\frac {x + 1}{x - 1} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.13, size = 26, normalized size = 1.37 \begin {gather*} \frac {\ln \left (x^2-1\right )}{2}+x\,\left (\frac {\ln \left (x+1\right )}{2}-\frac {\ln \left (1-x\right )}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(1/x),x)

[Out]

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

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