3.256 \(\int \frac{1}{(1-a^2 x^2) \tanh ^{-1}(a x)^3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0257797, 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} \[ -\frac{1}{2 a \tanh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - a^2*x^2)*ArcTanh[a*x]^3),x]

[Out]

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

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{1}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a \tanh ^{-1}(a x)^2}\\ \end{align*}

Mathematica [A]  time = 0.005255, size = 13, normalized size = 1. \[ -\frac{1}{2 a \tanh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - a^2*x^2)*ArcTanh[a*x]^3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*x^2+1)/arctanh(a*x)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.973626, size = 57, normalized size = 4.38 \begin{align*} -\frac{2}{a \log \left (a x + 1\right )^{2} - 2 \, a \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + a \log \left (-a x + 1\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)/arctanh(a*x)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.24365, size = 49, normalized size = 3.77 \begin{align*} -\frac{2}{a \log \left (-\frac{a x + 1}{a x - 1}\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)/arctanh(a*x)^3,x, algorithm="fricas")

[Out]

-2/(a*log(-(a*x + 1)/(a*x - 1))^2)

________________________________________________________________________________________

Sympy [A]  time = 1.4995, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{2 a \operatorname{atanh}^{2}{\left (a x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*x**2+1)/atanh(a*x)**3,x)

[Out]

-1/(2*a*atanh(a*x)**2)

________________________________________________________________________________________

Giac [A]  time = 1.19174, size = 30, normalized size = 2.31 \begin{align*} -\frac{2}{a \log \left (-\frac{a x + 1}{a x - 1}\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)/arctanh(a*x)^3,x, algorithm="giac")

[Out]

-2/(a*log(-(a*x + 1)/(a*x - 1))^2)