3.278 \(\int \frac{\tanh ^{-1}(\coth (a+b x))}{x} \, dx\)

Optimal. Leaf size=21 \[ b x-\log (x) \left (b x-\tanh ^{-1}(\coth (a+b x))\right ) \]

[Out]

b*x - (b*x - ArcTanh[Coth[a + b*x]])*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0303547, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2158, 29} \[ b x-\log (x) \left (b x-\tanh ^{-1}(\coth (a+b x))\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Coth[a + b*x]]/x,x]

[Out]

b*x - (b*x - ArcTanh[Coth[a + b*x]])*Log[x]

Rule 2158

Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(b*x)/a, x] - Dist[(b*u
- a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0161014, size = 19, normalized size = 0.9 \[ \log (x) \left (\tanh ^{-1}(\coth (a+b x))-b x\right )+b x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Coth[a + b*x]]/x,x]

[Out]

b*x + (-(b*x) + ArcTanh[Coth[a + b*x]])*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 21, normalized size = 1. \begin{align*} \ln \left ( x \right ){\it Artanh} \left ({\rm coth} \left (bx+a\right ) \right ) -\ln \left ( x \right ) xb+bx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(coth(b*x+a))/x,x)

[Out]

ln(x)*arctanh(coth(b*x+a))-ln(x)*x*b+b*x

________________________________________________________________________________________

Maxima [A]  time = 0.95269, size = 46, normalized size = 2.19 \begin{align*} -b{\left (x + \frac{a}{b}\right )} \log \left (x\right ) + b{\left (x + \frac{a \log \left (x\right )}{b}\right )} + \operatorname{artanh}\left (\coth \left (b x + a\right )\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(coth(b*x+a))/x,x, algorithm="maxima")

[Out]

-b*(x + a/b)*log(x) + b*(x + a*log(x)/b) + arctanh(coth(b*x + a))*log(x)

________________________________________________________________________________________

Fricas [A]  time = 2.13451, size = 22, normalized size = 1.05 \begin{align*} b x + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(coth(b*x+a))/x,x, algorithm="fricas")

[Out]

b*x + a*log(x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(coth(b*x+a))/x,x)

[Out]

Integral(atanh(coth(a + b*x))/x, x)

________________________________________________________________________________________

Giac [C]  time = 1.16903, size = 20, normalized size = 0.95 \begin{align*} b x + \frac{1}{2} \,{\left (i \, \pi + 2 \, a\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(coth(b*x+a))/x,x, algorithm="giac")

[Out]

b*x + 1/2*(I*pi + 2*a)*log(x)