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

Optimal. Leaf size=12 \[ \frac{\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b} \]

[Out]

Log[ArcTanh[Tanh[a + b*x]]]/b

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Rubi [A]  time = 0.0043055, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2157, 29} \[ \frac{\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^(-1),x]

[Out]

Log[ArcTanh[Tanh[a + b*x]]]/b

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 29

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

Rubi steps

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

Mathematica [A]  time = 0.0466461, size = 12, normalized size = 1. \[ \frac{\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^(-1),x]

[Out]

Log[ArcTanh[Tanh[a + b*x]]]/b

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Maple [A]  time = 0.027, size = 13, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctanh(tanh(b*x+a)),x)

[Out]

ln(arctanh(tanh(b*x+a)))/b

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Maxima [A]  time = 1.46089, size = 18, normalized size = 1.5 \begin{align*} \frac{\log \left (-b x - a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a)),x, algorithm="maxima")

[Out]

log(-b*x - a)/b

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Fricas [A]  time = 1.45088, size = 22, normalized size = 1.83 \begin{align*} \frac{\log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a)),x, algorithm="fricas")

[Out]

log(b*x + a)/b

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Sympy [A]  time = 9.01365, size = 17, normalized size = 1.42 \begin{align*} \begin{cases} \frac{\log{\left (\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\operatorname{atanh}{\left (\tanh{\left (a \right )} \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/atanh(tanh(b*x+a)),x)

[Out]

Piecewise((log(atanh(tanh(a + b*x)))/b, Ne(b, 0)), (x/atanh(tanh(a)), True))

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Giac [A]  time = 1.15638, size = 15, normalized size = 1.25 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a)),x, algorithm="giac")

[Out]

log(abs(b*x + a))/b