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

Optimal. Leaf size=44 \[ \frac{\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}-\frac{\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))} \]

[Out]

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

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Rubi [A]  time = 0.0265642, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2160, 2157, 29} \[ \frac{\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}-\frac{\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*ArcTanh[Tanh[a + b*x]]),x]

[Out]

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

Rule 2160

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

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

Mathematica [A]  time = 0.0188116, size = 29, normalized size = 0.66 \[ \frac{\log \left (\tanh ^{-1}(\tanh (a+b x))\right )-\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, size = 43, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}-{\frac{\ln \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) }{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(arctanh(tanh(b*x+a))-b*x)*ln(x)-1/(arctanh(tanh(b*x+a))-b*x)*ln(arctanh(tanh(b*x+a)))

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Maxima [A]  time = 1.77811, size = 24, normalized size = 0.55 \begin{align*} -\frac{\log \left (b x + a\right )}{a} + \frac{\log \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(b*x + a)/a + log(x)/a

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Fricas [A]  time = 1.55827, size = 38, normalized size = 0.86 \begin{align*} -\frac{\log \left (b x + a\right ) - \log \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(b*x + a) - log(x))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15711, size = 27, normalized size = 0.61 \begin{align*} -\frac{\log \left ({\left | b x + a \right |}\right )}{a} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(b*x + a))/a + log(abs(x))/a