3.203 \(\int x^4 (1-a^2 x^2)^2 \tanh ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=202 \[ -\frac{8 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{315 a^5}+\frac{a^2 x^7}{252}-\frac{67 x^3}{11340 a^2}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{29 x}{3780 a^4}+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}-\frac{16 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{315 a^5}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{23 x^5}{3780} \]

[Out]

(29*x)/(3780*a^4) - (67*x^3)/(11340*a^2) - (23*x^5)/3780 + (a^2*x^7)/252 - (29*ArcTanh[a*x])/(3780*a^5) + (8*x
^2*ArcTanh[a*x])/(315*a^3) + (4*x^4*ArcTanh[a*x])/(315*a) - (11*a*x^6*ArcTanh[a*x])/189 + (a^3*x^8*ArcTanh[a*x
])/36 + (8*ArcTanh[a*x]^2)/(315*a^5) + (x^5*ArcTanh[a*x]^2)/5 - (2*a^2*x^7*ArcTanh[a*x]^2)/7 + (a^4*x^9*ArcTan
h[a*x]^2)/9 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(315*a^5) - (8*PolyLog[2, 1 - 2/(1 - a*x)])/(315*a^5)

________________________________________________________________________________________

Rubi [A]  time = 1.02455, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 59, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6012, 5916, 5980, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac{8 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{315 a^5}+\frac{a^2 x^7}{252}-\frac{67 x^3}{11340 a^2}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{29 x}{3780 a^4}+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}-\frac{16 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{315 a^5}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{23 x^5}{3780} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]

[Out]

(29*x)/(3780*a^4) - (67*x^3)/(11340*a^2) - (23*x^5)/3780 + (a^2*x^7)/252 - (29*ArcTanh[a*x])/(3780*a^5) + (8*x
^2*ArcTanh[a*x])/(315*a^3) + (4*x^4*ArcTanh[a*x])/(315*a) - (11*a*x^6*ArcTanh[a*x])/189 + (a^3*x^8*ArcTanh[a*x
])/36 + (8*ArcTanh[a*x]^2)/(315*a^5) + (x^5*ArcTanh[a*x]^2)/5 - (2*a^2*x^7*ArcTanh[a*x]^2)/7 + (a^4*x^9*ArcTan
h[a*x]^2)/9 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(315*a^5) - (8*PolyLog[2, 1 - 2/(1 - a*x)])/(315*a^5)

Rule 6012

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[E
xpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[
c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 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 5980

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 5984

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

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int x^4 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=\int \left (x^4 \tanh ^{-1}(a x)^2-2 a^2 x^6 \tanh ^{-1}(a x)^2+a^4 x^8 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^6 \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^8 \tanh ^{-1}(a x)^2 \, dx+\int x^4 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{1}{5} (2 a) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{7} \left (4 a^3\right ) \int \frac{x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac{1}{9} \left (2 a^5\right ) \int \frac{x^9 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{2 \int x^3 \tanh ^{-1}(a x) \, dx}{5 a}-\frac{2 \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a}-\frac{1}{7} (4 a) \int x^5 \tanh ^{-1}(a x) \, dx+\frac{1}{7} (4 a) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{9} \left (2 a^3\right ) \int x^7 \tanh ^{-1}(a x) \, dx-\frac{1}{9} \left (2 a^3\right ) \int \frac{x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{x^4 \tanh ^{-1}(a x)}{10 a}-\frac{2}{21} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{1}{10} \int \frac{x^4}{1-a^2 x^2} \, dx+\frac{2 \int x \tanh ^{-1}(a x) \, dx}{5 a^3}-\frac{2 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^3}-\frac{4 \int x^3 \tanh ^{-1}(a x) \, dx}{7 a}+\frac{4 \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}+\frac{1}{9} (2 a) \int x^5 \tanh ^{-1}(a x) \, dx-\frac{1}{9} (2 a) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{21} \left (2 a^2\right ) \int \frac{x^6}{1-a^2 x^2} \, dx-\frac{1}{36} a^4 \int \frac{x^8}{1-a^2 x^2} \, dx\\ &=\frac{x^2 \tanh ^{-1}(a x)}{5 a^3}-\frac{3 x^4 \tanh ^{-1}(a x)}{70 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{\tanh ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{1}{10} \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx+\frac{1}{7} \int \frac{x^4}{1-a^2 x^2} \, dx-\frac{2 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac{4 \int x \tanh ^{-1}(a x) \, dx}{7 a^3}+\frac{4 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a^3}-\frac{\int \frac{x^2}{1-a^2 x^2} \, dx}{5 a^2}+\frac{2 \int x^3 \tanh ^{-1}(a x) \, dx}{9 a}-\frac{2 \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{9 a}-\frac{1}{27} a^2 \int \frac{x^6}{1-a^2 x^2} \, dx+\frac{1}{21} \left (2 a^2\right ) \int \left (-\frac{1}{a^6}-\frac{x^2}{a^4}-\frac{x^4}{a^2}+\frac{1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx-\frac{1}{36} a^4 \int \left (-\frac{1}{a^8}-\frac{x^2}{a^6}-\frac{x^4}{a^4}-\frac{x^6}{a^2}+\frac{1}{a^8 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac{293 x}{1260 a^4}+\frac{41 x^3}{3780 a^2}-\frac{17 x^5}{1260}+\frac{a^2 x^7}{252}-\frac{3 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac{3 \tanh ^{-1}(a x)^2}{35 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{2 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{5 a^5}-\frac{1}{18} \int \frac{x^4}{1-a^2 x^2} \, dx+\frac{1}{7} \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{36 a^4}+\frac{2 \int \frac{1}{1-a^2 x^2} \, dx}{21 a^4}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{10 a^4}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{5 a^4}+\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}+\frac{4 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^4}+\frac{2 \int x \tanh ^{-1}(a x) \, dx}{9 a^3}-\frac{2 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{9 a^3}+\frac{2 \int \frac{x^2}{1-a^2 x^2} \, dx}{7 a^2}-\frac{1}{27} a^2 \int \left (-\frac{1}{a^6}-\frac{x^2}{a^4}-\frac{x^4}{a^2}+\frac{1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=-\frac{601 x}{3780 a^4}-\frac{277 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}-\frac{293 \tanh ^{-1}(a x)}{1260 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{6 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{35 a^5}-\frac{1}{18} \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{5 a^5}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{27 a^4}+\frac{\int \frac{1}{1-a^2 x^2} \, dx}{7 a^4}-\frac{2 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{9 a^4}+\frac{2 \int \frac{1}{1-a^2 x^2} \, dx}{7 a^4}-\frac{4 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^4}-\frac{\int \frac{x^2}{1-a^2 x^2} \, dx}{9 a^2}\\ &=\frac{29 x}{3780 a^4}-\frac{67 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}+\frac{601 \tanh ^{-1}(a x)}{3780 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{315 a^5}-\frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{4 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{7 a^5}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{18 a^4}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{9 a^4}+\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{9 a^4}\\ &=\frac{29 x}{3780 a^4}-\frac{67 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{315 a^5}+\frac{3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{35 a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{9 a^5}\\ &=\frac{29 x}{3780 a^4}-\frac{67 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{315 a^5}-\frac{8 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{315 a^5}\\ \end{align*}

Mathematica [A]  time = 1.8055, size = 138, normalized size = 0.68 \[ \frac{288 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+a x \left (45 a^6 x^6-69 a^4 x^4-67 a^2 x^2+87\right )+36 \left (35 a^9 x^9-90 a^7 x^7+63 a^5 x^5-8\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (105 a^8 x^8-220 a^6 x^6+48 a^4 x^4+96 a^2 x^2-192 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-29\right )}{11340 a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]

[Out]

(a*x*(87 - 67*a^2*x^2 - 69*a^4*x^4 + 45*a^6*x^6) + 36*(-8 + 63*a^5*x^5 - 90*a^7*x^7 + 35*a^9*x^9)*ArcTanh[a*x]
^2 + 3*ArcTanh[a*x]*(-29 + 96*a^2*x^2 + 48*a^4*x^4 - 220*a^6*x^6 + 105*a^8*x^8 - 192*Log[1 + E^(-2*ArcTanh[a*x
])]) + 288*PolyLog[2, -E^(-2*ArcTanh[a*x])])/(11340*a^5)

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Maple [A]  time = 0.049, size = 259, normalized size = 1.3 \begin{align*}{\frac{{a}^{4}{x}^{9} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{9}}-{\frac{2\,{a}^{2}{x}^{7} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{7}}+{\frac{{x}^{5} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{{a}^{3}{x}^{8}{\it Artanh} \left ( ax \right ) }{36}}-{\frac{11\,a{x}^{6}{\it Artanh} \left ( ax \right ) }{189}}+{\frac{4\,{x}^{4}{\it Artanh} \left ( ax \right ) }{315\,a}}+{\frac{8\,{x}^{2}{\it Artanh} \left ( ax \right ) }{315\,{a}^{3}}}+{\frac{8\,{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{315\,{a}^{5}}}+{\frac{8\,{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{315\,{a}^{5}}}+{\frac{{a}^{2}{x}^{7}}{252}}-{\frac{23\,{x}^{5}}{3780}}-{\frac{67\,{x}^{3}}{11340\,{a}^{2}}}+{\frac{29\,x}{3780\,{a}^{4}}}+{\frac{29\,\ln \left ( ax-1 \right ) }{7560\,{a}^{5}}}-{\frac{29\,\ln \left ( ax+1 \right ) }{7560\,{a}^{5}}}+{\frac{2\, \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{315\,{a}^{5}}}-{\frac{8}{315\,{a}^{5}}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,\ln \left ( ax-1 \right ) }{315\,{a}^{5}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4}{315\,{a}^{5}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,\ln \left ( ax+1 \right ) }{315\,{a}^{5}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{2\, \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{315\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x)

[Out]

1/9*a^4*x^9*arctanh(a*x)^2-2/7*a^2*x^7*arctanh(a*x)^2+1/5*x^5*arctanh(a*x)^2+1/36*a^3*x^8*arctanh(a*x)-11/189*
a*x^6*arctanh(a*x)+4/315*x^4*arctanh(a*x)/a+8/315*x^2*arctanh(a*x)/a^3+8/315/a^5*arctanh(a*x)*ln(a*x-1)+8/315/
a^5*arctanh(a*x)*ln(a*x+1)+1/252*a^2*x^7-23/3780*x^5-67/11340*x^3/a^2+29/3780*x/a^4+29/7560/a^5*ln(a*x-1)-29/7
560/a^5*ln(a*x+1)+2/315/a^5*ln(a*x-1)^2-8/315/a^5*dilog(1/2+1/2*a*x)-4/315/a^5*ln(a*x-1)*ln(1/2+1/2*a*x)-4/315
/a^5*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)+4/315/a^5*ln(-1/2*a*x+1/2)*ln(a*x+1)-2/315/a^5*ln(a*x+1)^2

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Maxima [A]  time = 0.981327, size = 289, normalized size = 1.43 \begin{align*} \frac{1}{22680} \, a^{2}{\left (\frac{90 \, a^{7} x^{7} - 138 \, a^{5} x^{5} - 134 \, a^{3} x^{3} + 174 \, a x - 144 \, \log \left (a x + 1\right )^{2} + 288 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 144 \, \log \left (a x - 1\right )^{2} + 87 \, \log \left (a x - 1\right )}{a^{7}} - \frac{576 \,{\left (\log \left (a x - 1\right ) \log \left (\frac{1}{2} \, a x + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, a x + \frac{1}{2}\right )\right )}}{a^{7}} - \frac{87 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac{1}{3780} \, a{\left (\frac{105 \, a^{6} x^{8} - 220 \, a^{4} x^{6} + 48 \, a^{2} x^{4} + 96 \, x^{2}}{a^{4}} + \frac{96 \, \log \left (a x + 1\right )}{a^{6}} + \frac{96 \, \log \left (a x - 1\right )}{a^{6}}\right )} \operatorname{artanh}\left (a x\right ) + \frac{1}{315} \,{\left (35 \, a^{4} x^{9} - 90 \, a^{2} x^{7} + 63 \, x^{5}\right )} \operatorname{artanh}\left (a x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="maxima")

[Out]

1/22680*a^2*((90*a^7*x^7 - 138*a^5*x^5 - 134*a^3*x^3 + 174*a*x - 144*log(a*x + 1)^2 + 288*log(a*x + 1)*log(a*x
- 1) + 144*log(a*x - 1)^2 + 87*log(a*x - 1))/a^7 - 576*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/
2))/a^7 - 87*log(a*x + 1)/a^7) + 1/3780*a*((105*a^6*x^8 - 220*a^4*x^6 + 48*a^2*x^4 + 96*x^2)/a^4 + 96*log(a*x
+ 1)/a^6 + 96*log(a*x - 1)/a^6)*arctanh(a*x) + 1/315*(35*a^4*x^9 - 90*a^2*x^7 + 63*x^5)*arctanh(a*x)^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} x^{8} - 2 \, a^{2} x^{6} + x^{4}\right )} \operatorname{artanh}\left (a x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="fricas")

[Out]

integral((a^4*x^8 - 2*a^2*x^6 + x^4)*arctanh(a*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{2}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-a**2*x**2+1)**2*atanh(a*x)**2,x)

[Out]

Integral(x**4*(a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} x^{2} - 1\right )}^{2} x^{4} \operatorname{artanh}\left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="giac")

[Out]

integrate((a^2*x^2 - 1)^2*x^4*arctanh(a*x)^2, x)