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

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

[Out]

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

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Rubi [A]  time = 0.0309233, 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} \[ b x-\log (x) \left (b x-\coth ^{-1}(\coth (a+b x))\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Coth[a + b*x]]/x,x]

[Out]

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

Mathematica [A]  time = 0.0166107, size = 19, normalized size = 0.9 \[ \log (x) \left (\coth ^{-1}(\coth (a+b x))-b x\right )+b x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[Coth[a + b*x]]/x,x]

[Out]

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

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Maple [A]  time = 0.073, size = 27, normalized size = 1.3 \begin{align*} bx+a\ln \left ( x \right ) +\ln \left ( x \right ) \left ({\rm arccoth} \left ({\rm coth} \left (bx+a\right )\right )-bx-a \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.944969, size = 11, normalized size = 0.52 \begin{align*} b x + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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Fricas [A]  time = 1.48964, 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(arccoth(coth(b*x+a))/x,x, algorithm="fricas")

[Out]

b*x + a*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(coth(a + b*x))/x, x)

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Giac [A]  time = 1.13719, size = 12, normalized size = 0.57 \begin{align*} b x + a \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(abs(x))