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

Optimal. Leaf size=13 \[ \frac{\tanh ^{-1}(a x)^2}{2 a} \]

[Out]

ArcTanh[a*x]^2/(2*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0157754, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {5948} \[ \frac{\tanh ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(1 - a^2*x^2),x]

[Out]

ArcTanh[a*x]^2/(2*a)

Rule 5948

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

Rubi steps

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

Mathematica [A]  time = 0.0042339, size = 13, normalized size = 1. \[ \frac{\tanh ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[a*x]^2/(2*a)

________________________________________________________________________________________

Maple [A]  time = 0.025, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctanh(a*x)^2/a

________________________________________________________________________________________

Maxima [B]  time = 0.975797, size = 88, normalized size = 6.77 \begin{align*} \frac{1}{2} \,{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \operatorname{artanh}\left (a x\right ) - \frac{\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2}}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(log(a*x + 1)/a - log(a*x - 1)/a)*arctanh(a*x) - 1/8*(log(a*x + 1)^2 - 2*log(a*x + 1)*log(a*x - 1) + log(a
*x - 1)^2)/a

________________________________________________________________________________________

Fricas [A]  time = 1.87087, size = 47, normalized size = 3.62 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{2}}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(-(a*x + 1)/(a*x - 1))^2/a

________________________________________________________________________________________

Sympy [A]  time = 1.45779, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{2 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((atanh(a*x)**2/(2*a), Ne(a, 0)), (0, True))

________________________________________________________________________________________

Giac [A]  time = 1.13503, size = 30, normalized size = 2.31 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{2}}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(-(a*x + 1)/(a*x - 1))^2/a