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

Optimal. Leaf size=214 \[ \frac{245 x}{1152 \left (1-a^2 x^2\right )}+\frac{65 x}{1728 \left (1-a^2 x^2\right )^2}+\frac{x}{108 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}-\frac{5 \tanh ^{-1}(a x)}{48 a \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac{5 \tanh ^{-1}(a x)^3}{48 a}+\frac{245 \tanh ^{-1}(a x)}{1152 a} \]

[Out]

x/(108*(1 - a^2*x^2)^3) + (65*x)/(1728*(1 - a^2*x^2)^2) + (245*x)/(1152*(1 - a^2*x^2)) + (245*ArcTanh[a*x])/(1
152*a) - ArcTanh[a*x]/(18*a*(1 - a^2*x^2)^3) - (5*ArcTanh[a*x])/(48*a*(1 - a^2*x^2)^2) - (5*ArcTanh[a*x])/(16*
a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x]^2)/(24*(1 - a^2*x^2)^2) + (5*x*A
rcTanh[a*x]^2)/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^3)/(48*a)

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Rubi [A]  time = 0.168887, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5964, 5956, 5994, 199, 206} \[ \frac{245 x}{1152 \left (1-a^2 x^2\right )}+\frac{65 x}{1728 \left (1-a^2 x^2\right )^2}+\frac{x}{108 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}-\frac{5 \tanh ^{-1}(a x)}{48 a \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac{5 \tanh ^{-1}(a x)^3}{48 a}+\frac{245 \tanh ^{-1}(a x)}{1152 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^4,x]

[Out]

x/(108*(1 - a^2*x^2)^3) + (65*x)/(1728*(1 - a^2*x^2)^2) + (245*x)/(1152*(1 - a^2*x^2)) + (245*ArcTanh[a*x])/(1
152*a) - ArcTanh[a*x]/(18*a*(1 - a^2*x^2)^3) - (5*ArcTanh[a*x])/(48*a*(1 - a^2*x^2)^2) - (5*ArcTanh[a*x])/(16*
a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x]^2)/(24*(1 - a^2*x^2)^2) + (5*x*A
rcTanh[a*x]^2)/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^3)/(48*a)

Rule 5964

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^(
q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q
 + 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*
x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c
, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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])

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx &=-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}+\frac{1}{18} \int \frac{1}{\left (1-a^2 x^2\right )^4} \, dx+\frac{5}{6} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx\\ &=\frac{x}{108 \left (1-a^2 x^2\right )^3}-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)}{48 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac{5}{108} \int \frac{1}{\left (1-a^2 x^2\right )^3} \, dx+\frac{5}{48} \int \frac{1}{\left (1-a^2 x^2\right )^3} \, dx+\frac{5}{8} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{x}{108 \left (1-a^2 x^2\right )^3}+\frac{65 x}{1728 \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)}{48 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^3}{48 a}+\frac{5}{144} \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx+\frac{5}{64} \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{8} (5 a) \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{x}{108 \left (1-a^2 x^2\right )^3}+\frac{65 x}{1728 \left (1-a^2 x^2\right )^2}+\frac{65 x}{1152 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)}{48 a \left (1-a^2 x^2\right )^2}-\frac{5 \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^3}{48 a}+\frac{5}{288} \int \frac{1}{1-a^2 x^2} \, dx+\frac{5}{128} \int \frac{1}{1-a^2 x^2} \, dx+\frac{5}{16} \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{x}{108 \left (1-a^2 x^2\right )^3}+\frac{65 x}{1728 \left (1-a^2 x^2\right )^2}+\frac{245 x}{1152 \left (1-a^2 x^2\right )}+\frac{65 \tanh ^{-1}(a x)}{1152 a}-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)}{48 a \left (1-a^2 x^2\right )^2}-\frac{5 \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^3}{48 a}+\frac{5}{32} \int \frac{1}{1-a^2 x^2} \, dx\\ &=\frac{x}{108 \left (1-a^2 x^2\right )^3}+\frac{65 x}{1728 \left (1-a^2 x^2\right )^2}+\frac{245 x}{1152 \left (1-a^2 x^2\right )}+\frac{245 \tanh ^{-1}(a x)}{1152 a}-\frac{\tanh ^{-1}(a x)}{18 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)}{48 a \left (1-a^2 x^2\right )^2}-\frac{5 \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^2}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^3}{48 a}\\ \end{align*}

Mathematica [A]  time = 0.288938, size = 157, normalized size = 0.73 \[ \frac{-\frac{1470 x}{a^2 x^2-1}+\frac{260 x}{\left (a^2 x^2-1\right )^2}-\frac{64 x}{\left (a^2 x^2-1\right )^3}-\frac{144 x \left (15 a^4 x^4-40 a^2 x^2+33\right ) \tanh ^{-1}(a x)^2}{\left (a^2 x^2-1\right )^3}+\frac{48 \left (45 a^4 x^4-105 a^2 x^2+68\right ) \tanh ^{-1}(a x)}{a \left (a^2 x^2-1\right )^3}-\frac{735 \log (1-a x)}{a}+\frac{735 \log (a x+1)}{a}+\frac{720 \tanh ^{-1}(a x)^3}{a}}{6912} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-64*x)/(-1 + a^2*x^2)^3 + (260*x)/(-1 + a^2*x^2)^2 - (1470*x)/(-1 + a^2*x^2) + (48*(68 - 105*a^2*x^2 + 45*a^
4*x^4)*ArcTanh[a*x])/(a*(-1 + a^2*x^2)^3) - (144*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^2)/(-1 + a^2*x^
2)^3 + (720*ArcTanh[a*x]^3)/a - (735*Log[1 - a*x])/a + (735*Log[1 + a*x])/a)/6912

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Maple [C]  time = 0.512, size = 3447, normalized size = 16.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*Pi*x^6-15/32*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*Pi*x^4+15/
32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*Pi*x^2-5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(I/((a*x+1
)^2/(-a^2*x^2+1)+1))^3+5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(
I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2+5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(I*(a
*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(
I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))-5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*c
sgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2+5/32*I*a^5/(a*x-1)^3/(
a*x+1)^3*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^6-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x
)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^6-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2
*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^6-5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(
-a^2*x^2+1)+1))^2*Pi*x^6-15/32*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*P
i*x^4+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^4+15/64*I*a^3/(a*x-1
)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^4+15/32*I*a^3/(a*
x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^4+15/32*I*a/(a*x-1)^3/(a*x+1)^3*arct
anh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^2-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*
x+1)^2/(a^2*x^2-1))^3*Pi*x^2-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1
)^2/(-a^2*x^2+1)+1))^3*Pi*x^2-15/32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^
2*Pi*x^2+15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-
1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^2-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*
x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^6-5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*ar
ctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*Pi*x^6+5/48*a^5/(a*x-1)^3/(a*x
+1)^3*arctanh(a*x)^3*x^6+245/1152*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)*x^6-5/16*a^3/(a*x-1)^3/(a*x+1)^3*arctan
h(a*x)^3*x^4-125/384*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)*x^4+5/16*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*x^2-35
/384*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)*x^2-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^
2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*Pi*x^6+5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*
x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^6+15/64*I*a^3/(a*x-1)^3/(a
*x+1)^3*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)
+1))^2*Pi*x^4+15/32*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-
a^2*x^2+1)^(1/2))*Pi*x^4+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(
a*x+1)/(-a^2*x^2+1)^(1/2))^2*Pi*x^4-15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1
))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^4-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x
)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^2-15/32
*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*Pi*
x^2-5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^
2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*cs
gn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*Pi*x^2-245/1152*a^4/(a*x-1)^3/(a*x+1)^3*x^5+2
5/54*a^2/(a*x-1)^3/(a*x+1)^3*x^3-5/48/a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3+299/1152/a/(a*x-1)^3/(a*x+1)^3*arct
anh(a*x)-1/48/a*arctanh(a*x)^2/(a*x+1)^3-15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*
x^2-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*Pi*x^4+5/6
4*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*cs
gn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*Pi*x^6+15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*csg
n(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^
2+1)+1))*Pi*x^2+1/16/a*arctanh(a*x)^2/(a*x-1)^2-5/32/a*arctanh(a*x)^2/(a*x-1)-5/32/a*arctanh(a*x)^2*ln(a*x-1)-
1/16/a*arctanh(a*x)^2/(a*x+1)^2-5/32/a*arctanh(a*x)^2/(a*x+1)+5/32/a*arctanh(a*x)^2*ln(a*x+1)-5/16/a*arctanh(a
*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-299/1152/(a*x-1)^3/(a*x+1)^3*x-1/48/a*arctanh(a*x)^2/(a*x-1)^3-5/32*I/a/(
a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2+5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-
1))^3+5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^
3+5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2

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Maxima [B]  time = 1.03495, size = 697, normalized size = 3.26 \begin{align*} -\frac{1}{96} \,{\left (\frac{2 \,{\left (15 \, a^{4} x^{5} - 40 \, a^{2} x^{3} + 33 \, x\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1} - \frac{15 \, \log \left (a x + 1\right )}{a} + \frac{15 \, \log \left (a x - 1\right )}{a}\right )} \operatorname{artanh}\left (a x\right )^{2} - \frac{{\left (1470 \, a^{5} x^{5} - 3200 \, a^{3} x^{3} - 90 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} + 270 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 90 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 1794 \, a x - 15 \,{\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right ) + 735 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{6912 \,{\left (a^{9} x^{6} - 3 \, a^{7} x^{4} + 3 \, a^{5} x^{2} - a^{3}\right )}} + \frac{{\left (180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} - 45 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} + 90 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 45 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 272\right )} a \operatorname{artanh}\left (a x\right )}{576 \,{\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(2*(15*a^4*x^5 - 40*a^2*x^3 + 33*x)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1) - 15*log(a*x + 1)/a + 15*log(a
*x - 1)/a)*arctanh(a*x)^2 - 1/6912*(1470*a^5*x^5 - 3200*a^3*x^3 - 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log
(a*x + 1)^3 + 270*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^2*log(a*x - 1) + 90*(a^6*x^6 - 3*a^4*x^4
+ 3*a^2*x^2 - 1)*log(a*x - 1)^3 + 1794*a*x - 15*(49*a^6*x^6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*
x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 - 49)*log(a*x + 1) + 735*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1
))*a^2/(a^9*x^6 - 3*a^7*x^4 + 3*a^5*x^2 - a^3) + 1/576*(180*a^4*x^4 - 420*a^2*x^2 - 45*(a^6*x^6 - 3*a^4*x^4 +
3*a^2*x^2 - 1)*log(a*x + 1)^2 + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - 45*(a^6*x
^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 + 272)*a*arctanh(a*x)/(a^8*x^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2)

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Fricas [A]  time = 2.03648, size = 412, normalized size = 1.93 \begin{align*} -\frac{1470 \, a^{5} x^{5} - 3200 \, a^{3} x^{3} - 90 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 36 \,{\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 1794 \, a x - 3 \,{\left (245 \, a^{6} x^{6} - 375 \, a^{4} x^{4} - 105 \, a^{2} x^{2} + 299\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{6912 \,{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6912*(1470*a^5*x^5 - 3200*a^3*x^3 - 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1))^3 +
36*(15*a^5*x^5 - 40*a^3*x^3 + 33*a*x)*log(-(a*x + 1)/(a*x - 1))^2 + 1794*a*x - 3*(245*a^6*x^6 - 375*a^4*x^4 -
105*a^2*x^2 + 299)*log(-(a*x + 1)/(a*x - 1)))/(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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