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

Optimal. Leaf size=40 \[ -\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}-\frac{a x^3}{12}+\frac{x}{4 a} \]

[Out]

x/(4*a) - (a*x^3)/12 - ((1 - a^2*x^2)^2*ArcTanh[a*x])/(4*a^2)

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Rubi [A]  time = 0.0215924, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5994} \[ -\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}-\frac{a x^3}{12}+\frac{x}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[x*(1 - a^2*x^2)*ArcTanh[a*x],x]

[Out]

x/(4*a) - (a*x^3)/12 - ((1 - a^2*x^2)^2*ArcTanh[a*x])/(4*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]

Rubi steps

\begin{align*} \int x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}+\frac{\int \left (1-a^2 x^2\right ) \, dx}{4 a}\\ &=\frac{x}{4 a}-\frac{a x^3}{12}-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0140596, size = 69, normalized size = 1.72 \[ -\frac{1}{4} a^2 x^4 \tanh ^{-1}(a x)+\frac{\log (1-a x)}{8 a^2}-\frac{\log (a x+1)}{8 a^2}-\frac{a x^3}{12}+\frac{1}{2} x^2 \tanh ^{-1}(a x)+\frac{x}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(4*a) - (a*x^3)/12 + (x^2*ArcTanh[a*x])/2 - (a^2*x^4*ArcTanh[a*x])/4 + Log[1 - a*x]/(8*a^2) - Log[1 + a*x]/(
8*a^2)

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Maple [A]  time = 0.027, size = 57, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{\it Artanh} \left ( ax \right ){x}^{4}}{4}}+{\frac{{\it Artanh} \left ( ax \right ){x}^{2}}{2}}-{\frac{{x}^{3}a}{12}}+{\frac{x}{4\,a}}+{\frac{\ln \left ( ax-1 \right ) }{8\,{a}^{2}}}-{\frac{\ln \left ( ax+1 \right ) }{8\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*arctanh(a*x)*x^4+1/2*arctanh(a*x)*x^2-1/12*x^3*a+1/4*x/a+1/8/a^2*ln(a*x-1)-1/8/a^2*ln(a*x+1)

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Maxima [A]  time = 0.965126, size = 50, normalized size = 1.25 \begin{align*} -\frac{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname{artanh}\left (a x\right )}{4 \, a^{2}} - \frac{a^{2} x^{3} - 3 \, x}{12 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a^2*x^2 - 1)^2*arctanh(a*x)/a^2 - 1/12*(a^2*x^3 - 3*x)/a

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Fricas [A]  time = 2.19443, size = 117, normalized size = 2.92 \begin{align*} -\frac{2 \, a^{3} x^{3} - 6 \, a x + 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{24 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(2*a^3*x^3 - 6*a*x + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)))/a^2

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Sympy [A]  time = 1.36611, size = 46, normalized size = 1.15 \begin{align*} \begin{cases} - \frac{a^{2} x^{4} \operatorname{atanh}{\left (a x \right )}}{4} - \frac{a x^{3}}{12} + \frac{x^{2} \operatorname{atanh}{\left (a x \right )}}{2} + \frac{x}{4 a} - \frac{\operatorname{atanh}{\left (a x \right )}}{4 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*x**4*atanh(a*x)/4 - a*x**3/12 + x**2*atanh(a*x)/2 + x/(4*a) - atanh(a*x)/(4*a**2), Ne(a, 0)),
(0, True))

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Giac [B]  time = 1.16326, size = 100, normalized size = 2.5 \begin{align*} -\frac{1}{8} \,{\left (a^{2} x^{4} - 2 \, x^{2}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - \frac{\log \left ({\left | a x + 1 \right |}\right )}{8 \, a^{2}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{8 \, a^{2}} - \frac{a^{7} x^{3} - 3 \, a^{5} x}{12 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(a^2*x^4 - 2*x^2)*log(-(a*x + 1)/(a*x - 1)) - 1/8*log(abs(a*x + 1))/a^2 + 1/8*log(abs(a*x - 1))/a^2 - 1/1
2*(a^7*x^3 - 3*a^5*x)/a^6