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

Optimal. Leaf size=175 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^2}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^2}+\frac{\sqrt{1-a^2 x^2}}{3 a^2}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{3 a^2} \]

[Out]

Sqrt[1 - a^2*x^2]/(3*a^2) + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a) - (2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*
ArcTanh[a*x])/(3*a^2) - ((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2)/(3*a^2) - ((I/3)*PolyLog[2, ((-I)*Sqrt[1 - a*x])/
Sqrt[1 + a*x]])/a^2 + ((I/3)*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^2

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Rubi [A]  time = 0.124585, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {5994, 5942, 5950} \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^2}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^2}+\frac{\sqrt{1-a^2 x^2}}{3 a^2}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{3 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - a^2*x^2]/(3*a^2) + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a) - (2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*
ArcTanh[a*x])/(3*a^2) - ((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2)/(3*a^2) - ((I/3)*PolyLog[2, ((-I)*Sqrt[1 - a*x])/
Sqrt[1 + a*x]])/a^2 + ((I/3)*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + 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 5942

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

Rule 5950

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

Rubi steps

\begin{align*} \int x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx &=-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{2 \int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx}{3 a}\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a^2}+\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{\int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a^2}+\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{3 a^2}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{3 a^2}+\frac{i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.393182, size = 135, normalized size = 0.77 \[ \frac{\sqrt{1-a^2 x^2} \left (-\frac{i \left (\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )}{\sqrt{1-a^2 x^2}}-\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+a x \tanh ^{-1}(a x)+1\right )}{3 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*(1 + a*x*ArcTanh[a*x] - (1 - a^2*x^2)*ArcTanh[a*x]^2 - (I*(ArcTanh[a*x]*(Log[1 - I/E^ArcTan
h[a*x]] - Log[1 + I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1
- a^2*x^2]))/(3*a^2)

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Maple [A]  time = 0.252, size = 175, normalized size = 1. \begin{align*}{\frac{{a}^{2}{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+ax{\it Artanh} \left ( ax \right ) - \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+1}{3\,{a}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{\frac{i}{3}}{\it Artanh} \left ( ax \right ) }{{a}^{2}}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{i}{3}}{\it Artanh} \left ( ax \right ) }{{a}^{2}}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{i}{3}}}{{a}^{2}}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{i}{3}}}{{a}^{2}}{\it dilog} \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/a^2*(-(a*x-1)*(a*x+1))^(1/2)*(a^2*x^2*arctanh(a*x)^2+a*x*arctanh(a*x)-arctanh(a*x)^2+1)-1/3*I*ln(1+I*(a*x+
1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^2+1/3*I*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^2-1/3*I*dilog(
1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^2+1/3*I*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-a^{2} x^{2} + 1} x \operatorname{artanh}\left (a x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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