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

Optimal. Leaf size=43 \[ \frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{x}{a \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0506343, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5994, 191} \[ \frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{x}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5994

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)
^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*e*(q + 1)), x] + Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan
h[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -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{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac{x}{a \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0357683, size = 27, normalized size = 0.63 \[ \frac{\tanh ^{-1}(a x)-a x}{a^2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.205, size = 66, normalized size = 1.5 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) -1}{2\,{a}^{2} \left ( ax-1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{\it Artanh} \left ( ax \right ) +1}{2\,{a}^{2} \left ( ax+1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955062, size = 53, normalized size = 1.23 \begin{align*} -\frac{x}{\sqrt{-a^{2} x^{2} + 1} a} + \frac{\operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.0054, size = 103, normalized size = 2.4 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x - \log \left (-\frac{a x + 1}{a x - 1}\right )\right )}}{2 \,{\left (a^{4} x^{2} - a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.23861, size = 82, normalized size = 1.91 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} a} + \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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