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

Optimal. Leaf size=32 \[ \frac{\sin ^{-1}(a x)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0469097, antiderivative size = 32, 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, 216} \[ \frac{\sin ^{-1}(a x)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]/a^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.031359, size = 29, normalized size = 0.91 \[ \frac{\sin ^{-1}(a x)-\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.222, size = 81, normalized size = 2.5 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) }{{a}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{i}{{a}^{2}}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+i \right ) }-{\frac{i}{{a}^{2}}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-i \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.44362, size = 57, normalized size = 1.78 \begin{align*} \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a} - \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.07488, size = 135, normalized size = 4.22 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \log \left (-\frac{a x + 1}{a x - 1}\right ) + 4 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, 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)^(1/2),x, algorithm="fricas")

[Out]

-1/2*(sqrt(-a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)) + 4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/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 )}}{\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(x*atanh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.21385, size = 63, normalized size = 1.97 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1} \log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \, 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)^(1/2),x, algorithm="giac")

[Out]

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