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

Optimal. Leaf size=42 \[ -\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0758943, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6008, 266, 63, 208} \[ -\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6008

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Sim
p[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(m + 1), Int[(f*x)
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d
+ e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.050512, size = 48, normalized size = 1.14 \[ -a \log \left (\sqrt{1-a^2 x^2}+1\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+a \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.231, size = 72, normalized size = 1.7 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) }{x}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-a\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +a\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.43981, size = 69, normalized size = 1.64 \begin{align*} -a \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.09887, size = 127, normalized size = 3.02 \begin{align*} \frac{2 \, a x \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1} \log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.27043, size = 150, normalized size = 3.57 \begin{align*} -\frac{1}{2} \, a \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) + \frac{1}{2} \, a \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right ) + \frac{1}{4} \,{\left (\frac{a^{4} x}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{x{\left | a \right |}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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