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

Optimal. Leaf size=13 \[ \frac{\tanh ^{-1}(a x)^3}{3 a} \]

[Out]

ArcTanh[a*x]^3/(3*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0261026, antiderivative size = 13, 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 = {5948} \[ \frac{\tanh ^{-1}(a x)^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^2/(1 - a^2*x^2),x]

[Out]

ArcTanh[a*x]^3/(3*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)^2}{1-a^2 x^2} \, dx &=\frac{\tanh ^{-1}(a x)^3}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0048607, size = 13, normalized size = 1. \[ \frac{\tanh ^{-1}(a x)^3}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[a*x]^3/(3*a)

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{3\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arctanh(a*x)^3/a

________________________________________________________________________________________

Maxima [B]  time = 0.975451, size = 171, normalized size = 13.15 \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 )^{2} - \frac{{\left (\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}\right )} \operatorname{artanh}\left (a x\right )}{4 \, a} + \frac{\log \left (a x + 1\right )^{3} - 3 \, \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 3 \, \log \left (a x + 1\right ) \log \left (a x - 1\right )^{2} - \log \left (a x - 1\right )^{3}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(log(a*x + 1)/a - log(a*x - 1)/a)*arctanh(a*x)^2 - 1/4*(log(a*x + 1)^2 - 2*log(a*x + 1)*log(a*x - 1) + log
(a*x - 1)^2)*arctanh(a*x)/a + 1/24*(log(a*x + 1)^3 - 3*log(a*x + 1)^2*log(a*x - 1) + 3*log(a*x + 1)*log(a*x -
1)^2 - log(a*x - 1)^3)/a

________________________________________________________________________________________

Fricas [A]  time = 2.27843, size = 49, normalized size = 3.77 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{3}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*log(-(a*x + 1)/(a*x - 1))^3/a

________________________________________________________________________________________

Sympy [A]  time = 2.23219, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{atanh}^{3}{\left (a x \right )}}{3 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)**2/(-a**2*x**2+1),x)

[Out]

Piecewise((atanh(a*x)**3/(3*a), Ne(a, 0)), (0, True))

________________________________________________________________________________________

Giac [A]  time = 1.163, size = 30, normalized size = 2.31 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{3}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*log(-(a*x + 1)/(a*x - 1))^3/a