3.7 \(\int \frac{\coth ^{-1}(a x)}{x} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{2} \text{PolyLog}\left (2,-\frac{1}{a x}\right )-\frac{1}{2} \text{PolyLog}\left (2,\frac{1}{a x}\right ) \]

[Out]

PolyLog[2, -(1/(a*x))]/2 - PolyLog[2, 1/(a*x)]/2

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Rubi [A]  time = 0.0097539, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5913} \[ \frac{1}{2} \text{PolyLog}\left (2,-\frac{1}{a x}\right )-\frac{1}{2} \text{PolyLog}\left (2,\frac{1}{a x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/x,x]

[Out]

PolyLog[2, -(1/(a*x))]/2 - PolyLog[2, 1/(a*x)]/2

Rule 5913

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b*PolyLog[2, -(c*x)^(-1)
])/2, x] - Simp[(b*PolyLog[2, 1/(c*x)])/2, x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}(a x)}{x} \, dx &=\frac{1}{2} \text{Li}_2\left (-\frac{1}{a x}\right )-\frac{1}{2} \text{Li}_2\left (\frac{1}{a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0081032, size = 26, normalized size = 0.93 \[ \frac{1}{2} \left (\text{PolyLog}\left (2,-\frac{1}{a x}\right )-\text{PolyLog}\left (2,\frac{1}{a x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]/x,x]

[Out]

(PolyLog[2, -(1/(a*x))] - PolyLog[2, 1/(a*x)])/2

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Maple [A]  time = 0.042, size = 37, normalized size = 1.3 \begin{align*} \ln \left ( ax \right ){\rm arccoth} \left (ax\right )-{\frac{{\it dilog} \left ( ax \right ) }{2}}-{\frac{{\it dilog} \left ( ax+1 \right ) }{2}}-{\frac{\ln \left ( ax \right ) \ln \left ( ax+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)/x,x)

[Out]

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

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Maxima [B]  time = 0.997304, size = 116, normalized size = 4.14 \begin{align*} -\frac{1}{2} \, a{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \log \left (x\right ) - \frac{1}{2} \, a{\left (\frac{\log \left (a x - 1\right ) \log \left (a x\right ) +{\rm Li}_2\left (-a x + 1\right )}{a} - \frac{\log \left (a x + 1\right ) \log \left (-a x\right ) +{\rm Li}_2\left (a x + 1\right )}{a}\right )} + \operatorname{arcoth}\left (a x\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x,x, algorithm="maxima")

[Out]

-1/2*a*(log(a*x + 1)/a - log(a*x - 1)/a)*log(x) - 1/2*a*((log(a*x - 1)*log(a*x) + dilog(-a*x + 1))/a - (log(a*
x + 1)*log(-a*x) + dilog(a*x + 1))/a) + arccoth(a*x)*log(x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (a x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x,x, algorithm="fricas")

[Out]

integral(arccoth(a*x)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}{\left (a x \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x)/x,x)

[Out]

Integral(acoth(a*x)/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/x,x, algorithm="giac")

[Out]

integrate(arccoth(a*x)/x, x)