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

Optimal. Leaf size=227 \[ -\frac {1}{105} a^4 x^5+\frac {19 a^2 x^3}{315}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}-\frac {16 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {16 \tanh ^{-1}(a x)^2}{35 a}-\frac {32 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a}-\frac {38 x}{105} \]

[Out]

-38/105*x+19/315*a^2*x^3-1/105*a^4*x^5+8/35*(-a^2*x^2+1)*arctanh(a*x)/a+3/35*(-a^2*x^2+1)^2*arctanh(a*x)/a+1/2
1*(-a^2*x^2+1)^3*arctanh(a*x)/a+16/35*arctanh(a*x)^2/a+16/35*x*arctanh(a*x)^2+8/35*x*(-a^2*x^2+1)*arctanh(a*x)
^2+6/35*x*(-a^2*x^2+1)^2*arctanh(a*x)^2+1/7*x*(-a^2*x^2+1)^3*arctanh(a*x)^2-32/35*arctanh(a*x)*ln(2/(-a*x+1))/
a-16/35*polylog(2,1-2/(-a*x+1))/a

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Rubi [A]  time = 0.17, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5944, 5910, 5984, 5918, 2402, 2315, 8, 194} \[ -\frac {16 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a}-\frac {1}{105} a^4 x^5+\frac {19 a^2 x^3}{315}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {16 \tanh ^{-1}(a x)^2}{35 a}-\frac {32 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a}-\frac {38 x}{105} \]

Antiderivative was successfully verified.

[In]

Int[(1 - a^2*x^2)^3*ArcTanh[a*x]^2,x]

[Out]

(-38*x)/105 + (19*a^2*x^3)/315 - (a^4*x^5)/105 + (8*(1 - a^2*x^2)*ArcTanh[a*x])/(35*a) + (3*(1 - a^2*x^2)^2*Ar
cTanh[a*x])/(35*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x])/(21*a) + (16*ArcTanh[a*x]^2)/(35*a) + (16*x*ArcTanh[a*x]^2
)/35 + (8*x*(1 - a^2*x^2)*ArcTanh[a*x]^2)/35 + (6*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/35 + (x*(1 - a^2*x^2)^3*Ar
cTanh[a*x]^2)/7 - (32*ArcTanh[a*x]*Log[2/(1 - a*x)])/(35*a) - (16*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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]

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 5910

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcTanh[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && 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 5944

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

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]

Rubi steps

\begin {align*} \int \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2 \, dx &=\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {1}{21} \int \left (1-a^2 x^2\right )^2 \, dx+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {1}{21} \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac {3}{35} \int \left (1-a^2 x^2\right ) \, dx+\frac {24}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac {2 x}{15}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {8 \int 1 \, dx}{35}+\frac {16}{35} \int \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {1}{35} (32 a) \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32}{35} \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}+\frac {32}{35} \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {32 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{35 a}\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {16 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}\\ \end {align*}

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Mathematica [A]  time = 1.22, size = 124, normalized size = 0.55 \[ -\frac {3 a^5 x^5-19 a^3 x^3+9 (a x-1)^4 \left (5 a^3 x^3+20 a^2 x^2+29 a x+16\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (5 a^6 x^6-24 a^4 x^4+57 a^2 x^2+96 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-38\right )-144 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+114 a x}{315 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/315*(114*a*x - 19*a^3*x^3 + 3*a^5*x^5 + 9*(-1 + a*x)^4*(16 + 29*a*x + 20*a^2*x^2 + 5*a^3*x^3)*ArcTanh[a*x]^
2 + 3*ArcTanh[a*x]*(-38 + 57*a^2*x^2 - 24*a^4*x^4 + 5*a^6*x^6 + 96*Log[1 + E^(-2*ArcTanh[a*x])]) - 144*PolyLog
[2, -E^(-2*ArcTanh[a*x])])/a

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*arctanh(a*x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 250, normalized size = 1.10 \[ -\frac {a^{6} \arctanh \left (a x \right )^{2} x^{7}}{7}+\frac {3 a^{4} \arctanh \left (a x \right )^{2} x^{5}}{5}-a^{2} \arctanh \left (a x \right )^{2} x^{3}+x \arctanh \left (a x \right )^{2}-\frac {a^{5} \arctanh \left (a x \right ) x^{6}}{21}+\frac {8 a^{3} \arctanh \left (a x \right ) x^{4}}{35}-\frac {19 a \arctanh \left (a x \right ) x^{2}}{35}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{35 a}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{35 a}+\frac {4 \ln \left (a x -1\right )^{2}}{35 a}-\frac {16 \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{35 a}-\frac {8 \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{35 a}-\frac {4 \ln \left (a x +1\right )^{2}}{35 a}+\frac {8 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{35 a}-\frac {8 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{35 a}-\frac {a^{4} x^{5}}{105}+\frac {19 x^{3} a^{2}}{315}-\frac {38 x}{105}-\frac {19 \ln \left (a x -1\right )}{105 a}+\frac {19 \ln \left (a x +1\right )}{105 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^6*arctanh(a*x)^2*x^7+3/5*a^4*arctanh(a*x)^2*x^5-a^2*arctanh(a*x)^2*x^3+x*arctanh(a*x)^2-1/21*a^5*arctan
h(a*x)*x^6+8/35*a^3*arctanh(a*x)*x^4-19/35*a*arctanh(a*x)*x^2+16/35/a*arctanh(a*x)*ln(a*x-1)+16/35/a*arctanh(a
*x)*ln(a*x+1)+4/35/a*ln(a*x-1)^2-16/35/a*dilog(1/2+1/2*a*x)-8/35/a*ln(a*x-1)*ln(1/2+1/2*a*x)-4/35/a*ln(a*x+1)^
2+8/35/a*ln(-1/2*a*x+1/2)*ln(a*x+1)-8/35/a*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)-1/105*a^4*x^5+19/315*x^3*a^2-38/10
5*x-19/105/a*ln(a*x-1)+19/105/a*ln(a*x+1)

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maxima [A]  time = 0.33, size = 199, normalized size = 0.88 \[ -\frac {1}{315} \, a^{2} {\left (\frac {3 \, a^{5} x^{5} - 19 \, a^{3} x^{3} + 114 \, a x + 36 \, \log \left (a x + 1\right )^{2} - 72 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 36 \, \log \left (a x - 1\right )^{2} + 57 \, \log \left (a x - 1\right )}{a^{3}} + \frac {144 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}} - \frac {57 \, \log \left (a x + 1\right )}{a^{3}}\right )} - \frac {1}{105} \, {\left (5 \, a^{4} x^{6} - 24 \, a^{2} x^{4} + 57 \, x^{2} - \frac {48 \, \log \left (a x + 1\right )}{a^{2}} - \frac {48 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 35 \, a^{2} x^{3} - 35 \, x\right )} \operatorname {artanh}\left (a x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*a^2*((3*a^5*x^5 - 19*a^3*x^3 + 114*a*x + 36*log(a*x + 1)^2 - 72*log(a*x + 1)*log(a*x - 1) - 36*log(a*x
- 1)^2 + 57*log(a*x - 1))/a^3 + 144*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^3 - 57*log(a*x
+ 1)/a^3) - 1/105*(5*a^4*x^6 - 24*a^2*x^4 + 57*x^2 - 48*log(a*x + 1)/a^2 - 48*log(a*x - 1)/a^2)*a*arctanh(a*x
) - 1/35*(5*a^6*x^7 - 21*a^4*x^5 + 35*a^2*x^3 - 35*x)*arctanh(a*x)^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ -\int {\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)^2*(a^2*x^2 - 1)^3,x)

[Out]

-int(atanh(a*x)^2*(a^2*x^2 - 1)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(3*a**2*x**2*atanh(a*x)**2, x) - Integral(-3*a**4*x**4*atanh(a*x)**2, x) - Integral(a**6*x**6*atanh(a
*x)**2, x) - Integral(-atanh(a*x)**2, x)

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