3.250 \(\int \frac{1}{(1-a^2 x^2) \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=9 \[ \frac{\log \left (\tanh ^{-1}(a x)\right )}{a} \]

[Out]

Log[ArcTanh[a*x]]/a

________________________________________________________________________________________

Rubi [A]  time = 0.0275559, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {5946} \[ \frac{\log \left (\tanh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - a^2*x^2)*ArcTanh[a*x]),x]

[Out]

Log[ArcTanh[a*x]]/a

Rule 5946

Int[1/(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[Log[RemoveContent[a + b*A
rcTanh[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \, dx &=\frac{\log \left (\tanh ^{-1}(a x)\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0286535, size = 9, normalized size = 1. \[ \frac{\log \left (\tanh ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - a^2*x^2)*ArcTanh[a*x]),x]

[Out]

Log[ArcTanh[a*x]]/a

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 10, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({\it Artanh} \left ( ax \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*x^2+1)/arctanh(a*x),x)

[Out]

ln(arctanh(a*x))/a

________________________________________________________________________________________

Maxima [B]  time = 0.955185, size = 28, normalized size = 3.11 \begin{align*} \frac{\log \left (-\log \left (a x + 1\right ) + \log \left (-a x + 1\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)/arctanh(a*x),x, algorithm="maxima")

[Out]

log(-log(a*x + 1) + log(-a*x + 1))/a

________________________________________________________________________________________

Fricas [B]  time = 2.26805, size = 46, normalized size = 5.11 \begin{align*} \frac{\log \left (\log \left (-\frac{a x + 1}{a x - 1}\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)/arctanh(a*x),x, algorithm="fricas")

[Out]

log(log(-(a*x + 1)/(a*x - 1)))/a

________________________________________________________________________________________

Sympy [A]  time = 0.891838, size = 7, normalized size = 0.78 \begin{align*} \frac{\log{\left (\operatorname{atanh}{\left (a x \right )} \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*x**2+1)/atanh(a*x),x)

[Out]

log(atanh(a*x))/a

________________________________________________________________________________________

Giac [B]  time = 1.17304, size = 28, normalized size = 3.11 \begin{align*} \frac{\log \left ({\left | \log \left (-\frac{a x + 1}{a x - 1}\right ) \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)/arctanh(a*x),x, algorithm="giac")

[Out]

log(abs(log(-(a*x + 1)/(a*x - 1))))/a