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

Optimal. Leaf size=205 \[ -\frac{5 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}+\frac{5 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}-\frac{\sqrt{1-a^2 x^2}}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{10 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{3 a^4} \]

[Out]

-Sqrt[1 - a^2*x^2]/(3*a^4) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^3) - (10*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*
x]]*ArcTanh[a*x])/(3*a^4) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]
^2)/(3*a^2) - (((5*I)/3)*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^4 + (((5*I)/3)*PolyLog[2, (I*Sqrt[1
- a*x])/Sqrt[1 + a*x]])/a^4

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Rubi [A]  time = 0.313618, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6016, 261, 5950, 5994} \[ -\frac{5 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}+\frac{5 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}-\frac{\sqrt{1-a^2 x^2}}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{10 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{3 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[1 - a^2*x^2]/(3*a^4) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^3) - (10*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*
x]]*ArcTanh[a*x])/(3*a^4) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]
^2)/(3*a^2) - (((5*I)/3)*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^4 + (((5*I)/3)*PolyLog[2, (I*Sqrt[1
- a*x])/Sqrt[1 + a*x]])/a^4

Rule 6016

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

Rule 261

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

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]

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 \frac{x^3 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{2 \int \frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}+\frac{2 \int \frac{x^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^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^3}+\frac{4 \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a^3}+\frac{\int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{10 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{3 a^4}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{5 i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{3 a^4}+\frac{5 i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{3 a^4}\\ \end{align*}

Mathematica [A]  time = 0.391785, size = 160, normalized size = 0.78 \[ \frac{\sqrt{1-a^2 x^2} \left (-\frac{5 i \left (\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt{1-a^2 x^2}}+\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2-\frac{5 i \tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt{1-a^2 x^2}}-3 \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x)-1\right )}{3 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*(-1 - a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + (1 - a^2*x^2)*ArcTanh[a*x]^2 - ((5*I)*ArcTanh[a
*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2] - ((5*I)*(PolyLog[2, (-I)/E^Arc
Tanh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(3*a^4)

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Maple [A]  time = 0.258, size = 175, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+ax{\it Artanh} \left ( ax \right ) +2\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+1}{3\,{a}^{4}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{\frac{5\,i}{3}}{\it Artanh} \left ( ax \right ) }{{a}^{4}}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{5\,i}{3}}{\it Artanh} \left ( ax \right ) }{{a}^{4}}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{5\,i}{3}}}{{a}^{4}}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{5\,i}{3}}}{{a}^{4}}{\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^3*arctanh(a*x)^2/(-a^2*x^2+1)^(1/2),x)

[Out]

-1/3/a^4*(-(a*x-1)*(a*x+1))^(1/2)*(a^2*x^2*arctanh(a*x)^2+a*x*arctanh(a*x)+2*arctanh(a*x)^2+1)-5/3*I*ln(1+I*(a
*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^4+5/3*I*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^4-5/3*I*dil
og(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^4+5/3*I*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{3} \operatorname{artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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