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(b*x+a)))*ln(x)

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 29

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

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*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.90 \[ \log (x) \left (\tanh ^{-1}(\coth (a+b x))-b x\right )+b x \]

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]

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fricas [A]  time = 0.71, size = 8, normalized size = 0.38 \[ b x + a \log \relax (x) \]

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)

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giac [C]  time = 0.14, size = 15, normalized size = 0.71 \[ b x + \frac {1}{2} \, {\left (i \, \pi + 2 \, a\right )} \log \relax (x) \]

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)

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maple [A]  time = 0.14, size = 21, normalized size = 1.00 \[ \ln \relax (x ) \arctanh \left (\coth \left (b x +a \right )\right )-\ln \relax (x ) x b +b x \]

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

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maxima [A]  time = 0.33, size = 34, normalized size = 1.62 \[ -b {\left (x + \frac {a}{b}\right )} \log \relax (x) + b {\left (x + \frac {a \log \relax (x)}{b}\right )} + \operatorname {artanh}\left (\coth \left (b x + a\right )\right ) \log \relax (x) \]

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)

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mupad [B]  time = 1.20, size = 59, normalized size = 2.81 \[ b\,x-\ln \relax (x)\,\left (\frac {\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}-\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}+b\,x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(coth(a + b*x))/x,x)

[Out]

b*x - log(x)*(log(-2/(exp(2*a)*exp(2*b*x) - 1))/2 - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1))/2 +
b*x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\left (\coth {\left (a + b x \right )} \right )}}{x}\, dx \]

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)

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