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

Optimal. Leaf size=11 \[ -\frac{1}{a \tanh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0247837, antiderivative size = 11, 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} \[ -\frac{1}{a \tanh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0047918, size = 11, normalized size = 1. \[ -\frac{1}{a \tanh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 12, normalized size = 1.1 \begin{align*} -{\frac{1}{a{\it Artanh} \left ( ax \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/arctanh(a*x)

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Maxima [B]  time = 0.970097, size = 31, normalized size = 2.82 \begin{align*} -\frac{2}{a \log \left (a x + 1\right ) - a \log \left (-a x + 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*log(a*x + 1) - a*log(-a*x + 1))

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Fricas [A]  time = 2.10922, size = 46, normalized size = 4.18 \begin{align*} -\frac{2}{a \log \left (-\frac{a x + 1}{a x - 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*log(-(a*x + 1)/(a*x - 1)))

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Sympy [A]  time = 1.21073, size = 8, normalized size = 0.73 \begin{align*} - \frac{1}{a \operatorname{atanh}{\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)**2,x)

[Out]

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

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Giac [A]  time = 1.14018, size = 30, normalized size = 2.73 \begin{align*} -\frac{2}{a \log \left (-\frac{a x + 1}{a x - 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*log(-(a*x + 1)/(a*x - 1)))