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

Optimal. Leaf size=191 \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3 a}{8 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{3}{8} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]

[Out]

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

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Rubi [A]  time = 0.435097, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6030, 5982, 5916, 5988, 5932, 5948, 6056, 6610, 5956, 5994, 261} \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{3 a}{8 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{3}{8} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6030

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

Rule 5982

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

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 5988

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

Rule 5932

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

Rule 5948

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

Rule 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u])
/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2
/(1 + c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rule 5956

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

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]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{1}{8} a \tanh ^{-1}(a x)^4+a^2 \int \frac{\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx-\frac{1}{2} \left (3 a^3\right ) \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^2} \, dx\\ &=-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^4+(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} \left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^4+(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx-\frac{1}{4} \left (3 a^3\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{3 a}{8 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-\left (6 a^2\right ) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{3 a}{8 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\left (3 a^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{3 a}{8 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^2-\frac{3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac{3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a \text{Li}_3\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}

Mathematica [C]  time = 0.341374, size = 144, normalized size = 0.75 \[ \frac{1}{16} a \left (48 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )-24 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x)^4-\frac{16 \tanh ^{-1}(a x)^3}{a x}-16 \tanh ^{-1}(a x)^3+48 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+4 \tanh ^{-1}(a x)^3 \sinh \left (2 \tanh ^{-1}(a x)\right )+6 \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )-6 \tanh ^{-1}(a x)^2 \cosh \left (2 \tanh ^{-1}(a x)\right )-3 \cosh \left (2 \tanh ^{-1}(a x)\right )+2 i \pi ^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*((2*I)*Pi^3 - 16*ArcTanh[a*x]^3 - (16*ArcTanh[a*x]^3)/(a*x) + 6*ArcTanh[a*x]^4 - 3*Cosh[2*ArcTanh[a*x]] - 6
*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]] + 48*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 48*ArcTanh[a*x]*PolyLog
[2, E^(2*ArcTanh[a*x])] - 24*PolyLog[3, E^(2*ArcTanh[a*x])] + 6*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]] + 4*ArcTanh[
a*x]^3*Sinh[2*ArcTanh[a*x]]))/16

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Maple [B]  time = 0.871, size = 442, normalized size = 2.3 \begin{align*} -6\,a{\it polylog} \left ( 3,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -6\,a{\it polylog} \left ( 3,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{3\,{a}^{2}x}{32\,ax+32}}+{\frac{3\,{a}^{2}x}{32\,ax-32}}+3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -{\frac{a \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{8\,ax-8}}-{\frac{a \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{8\,ax+8}}-{\frac{3\,a{\it Artanh} \left ( ax \right ) }{16\,ax-16}}-{\frac{3\,a{\it Artanh} \left ( ax \right ) }{16\,ax+16}}+{\frac{3\,a}{32\,ax-32}}-{\frac{3\,a}{32\,ax+32}}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{x}}-a \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}+{\frac{3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}{8}}+6\,a{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +6\,a{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{16\,ax-16}}-{\frac{3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{16\,ax+16}}-{\frac{3\,{\it Artanh} \left ( ax \right ){a}^{2}x}{16\,ax-16}}+{\frac{3\,{\it Artanh} \left ( ax \right ){a}^{2}x}{16\,ax+16}}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}x{a}^{2}}{8\,ax-8}}+{\frac{3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}x{a}^{2}}{16\,ax-16}}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}x{a}^{2}}{8\,ax+8}}+{\frac{3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}x{a}^{2}}{16\,ax+16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(a*x)^3/x-6*a*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*a*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a*arc
tanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3/32/(
a*x+1)*a^2*x+3/32*a^2*x/(a*x-1)+3/16*a*arctanh(a*x)^2/(a*x-1)-3/16*a*arctanh(a*x)^2/(a*x+1)-3/16*arctanh(a*x)/
(a*x-1)*a^2*x+3/16*arctanh(a*x)/(a*x+1)*a^2*x+3*a*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a*arctanh(
a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/8*a/(a*x-1)*arctanh(a*x)^3+3/32*a/(a*x-1)-3/32*a/(a*x+1)-a*arctanh(a
*x)^3+3/8*a*arctanh(a*x)^4-1/8*a/(a*x+1)*arctanh(a*x)^3-1/8/(a*x-1)*arctanh(a*x)^3*x*a^2+3/16/(a*x-1)*arctanh(
a*x)^2*x*a^2+1/8/(a*x+1)*arctanh(a*x)^3*x*a^2+3/16/(a*x+1)*arctanh(a*x)^2*x*a^2-3/16*a*arctanh(a*x)/(a*x-1)-3/
16*a*arctanh(a*x)/(a*x+1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{artanh}\left (a x\right )^{3}}{a^{4} x^{6} - 2 \, a^{2} x^{4} + x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctanh(a*x)^3/(a^4*x^6 - 2*a^2*x^4 + x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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