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

Optimal. Leaf size=63 \[ \frac{2 x}{\sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0396381, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5962, 191} \[ \frac{2 x}{\sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5962

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[(b*p*(a + b*ArcTa
nh[c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (Dist[b^2*p*(p - 1), Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2
)^(3/2), x], x] + Simp[(x*(a + b*ArcTanh[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ
[c^2*d + e, 0] && GtQ[p, 1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0406033, size = 38, normalized size = 0.6 \[ \frac{2 a x+a x \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.225, size = 49, normalized size = 0.8 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}ax+2\,ax-2\,{\it Artanh} \left ( ax \right ) }{a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955922, size = 77, normalized size = 1.22 \begin{align*} \frac{x \operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}} + \frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{2 \, \operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.28507, size = 150, normalized size = 2.38 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 8 \, a x - 4 \, \log \left (-\frac{a x + 1}{a x - 1}\right )\right )}}{4 \,{\left (a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-a^2*x^2 + 1)*(a*x*log(-(a*x + 1)/(a*x - 1))^2 + 8*a*x - 4*log(-(a*x + 1)/(a*x - 1)))/(a^3*x^2 - a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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