3.4.46 \(\int \frac {\tanh ^{-1}(a x)^2}{(1-a^2 x^2)^4} \, dx\) [346]

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

[Out]

1/108*x/(-a^2*x^2+1)^3+65/1728*x/(-a^2*x^2+1)^2+245/1152*x/(-a^2*x^2+1)+245/1152*arctanh(a*x)/a-1/18*arctanh(a
*x)/a/(-a^2*x^2+1)^3-5/48*arctanh(a*x)/a/(-a^2*x^2+1)^2-5/16*arctanh(a*x)/a/(-a^2*x^2+1)+1/6*x*arctanh(a*x)^2/
(-a^2*x^2+1)^3+5/24*x*arctanh(a*x)^2/(-a^2*x^2+1)^2+5/16*x*arctanh(a*x)^2/(-a^2*x^2+1)+5/48*arctanh(a*x)^3/a

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Rubi [A]
time = 0.12, antiderivative size = 214, normalized size of antiderivative = 1.00, 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 = {6111, 6103, 6141, 205, 212} \begin {gather*} \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} \end {gather*}

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 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

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

Rule 6103

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 6111

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 6141

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 {\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*}

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Mathematica [A]
time = 0.08, size = 157, normalized size = 0.73 \begin {gather*} \frac {-\frac {64 x}{\left (-1+a^2 x^2\right )^3}+\frac {260 x}{\left (-1+a^2 x^2\right )^2}-\frac {1470 x}{-1+a^2 x^2}+\frac {48 \left (68-105 a^2 x^2+45 a^4 x^4\right ) \tanh ^{-1}(a x)}{a \left (-1+a^2 x^2\right )^3}-\frac {144 x \left (33-40 a^2 x^2+15 a^4 x^4\right ) \tanh ^{-1}(a x)^2}{\left (-1+a^2 x^2\right )^3}+\frac {720 \tanh ^{-1}(a x)^3}{a}-\frac {735 \log (1-a x)}{a}+\frac {735 \log (1+a x)}{a}}{6912} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 10.11, size = 2716, normalized size = 12.69

method	result	size
risch	\(\frac {5 \ln \left (a x +1\right )^{3}}{384 a}-\frac {\left (15 a^{6} x^{6} \ln \left (-a x +1\right )+30 a^{5} x^{5}-45 x^{4} \ln \left (-a x +1\right ) a^{4}-80 a^{3} x^{3}+45 x^{2} \ln \left (-a x +1\right ) a^{2}+66 a x -15 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{384 \left (a^{2} x^{2}-1\right )^{3} a}+\frac {\left (45 a^{6} x^{6} \ln \left (-a x +1\right )^{2}+180 x^{5} \ln \left (-a x +1\right ) a^{5}-135 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+180 a^{4} x^{4}-480 a^{3} x^{3} \ln \left (-a x +1\right )+135 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-420 a^{2} x^{2}+396 a x \ln \left (-a x +1\right )-45 \ln \left (-a x +1\right )^{2}+272\right ) \ln \left (a x +1\right )}{1152 \left (a^{2} x^{2}-1\right )^{2} a \left (a x -1\right ) \left (a x +1\right )}+\frac {-90 a^{6} x^{6} \ln \left (-a x +1\right )^{3}+735 \ln \left (-a x -1\right ) a^{6} x^{6}-735 \ln \left (a x -1\right ) a^{6} x^{6}-540 a^{5} x^{5} \ln \left (-a x +1\right )^{2}+270 a^{4} x^{4} \ln \left (-a x +1\right )^{3}-1470 a^{5} x^{5}-2205 \ln \left (-a x -1\right ) a^{4} x^{4}+2205 \ln \left (a x -1\right ) a^{4} x^{4}-1080 x^{4} \ln \left (-a x +1\right ) a^{4}+1440 a^{3} x^{3} \ln \left (-a x +1\right )^{2}-270 a^{2} x^{2} \ln \left (-a x +1\right )^{3}+3200 a^{3} x^{3}+2205 \ln \left (-a x -1\right ) a^{2} x^{2}-2205 \ln \left (a x -1\right ) a^{2} x^{2}+2520 x^{2} \ln \left (-a x +1\right ) a^{2}-1188 a \ln \left (-a x +1\right )^{2} x +90 \ln \left (-a x +1\right )^{3}-1794 a x -735 \ln \left (-a x -1\right )+735 \ln \left (a x -1\right )-1632 \ln \left (-a x +1\right )}{6912 a \left (a^{2} x^{2}-1\right ) \left (a x +1\right )^{2} \left (a x -1\right )^{2}}\)	\(574\)
derivativedivides	\(\text {Expression too large to display}\)	\(2716\)
default	\(\text {Expression too large to display}\)	\(2716\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(-1/48*arctanh(a*x)^2/(a*x-1)^3+1/16*arctanh(a*x)^2/(a*x-1)^2-5/32*arctanh(a*x)^2/(a*x-1)-5/32*arctanh(a*x
)^2*ln(a*x-1)-1/48*arctanh(a*x)^2/(a*x+1)^3-1/16*arctanh(a*x)^2/(a*x+1)^2-5/32*arctanh(a*x)^2/(a*x+1)+5/32*arc
tanh(a*x)^2*ln(a*x+1)-5/16*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+1/3456*(1080*arctanh(a*x)^3*a^2*x^2-8
97*a*x-735*a^5*x^5-1125*a^4*x^4*arctanh(a*x)-360*arctanh(a*x)^3-315*a^2*x^2*arctanh(a*x)+897*arctanh(a*x)+1600
*a^3*x^3-1080*arctanh(a*x)^3*a^4*x^4+810*I*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*a^4*x^4-1620*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*a^4*x^4-810*I*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))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2
/(a^2*x^2-1))*a^4*x^4+810*I*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))*csgn(I/
((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*a^2*x^2+270*I*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))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*a^6*x^6-
540*I*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*a^6*x^6-270*I*Pi*ar
ctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*a^6
*x^6+270*I*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))^2*csgn(I*(a*x+1)^2/(a^2*
x^2-1))*a^6*x^6+810*I*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))*a^4
*x^4+1620*I*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*a^4*x^4+810*I
*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))^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1
))*a^4*x^4-810*I*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))^2*csgn(I*(a*x+1)^2
/(a^2*x^2-1))*a^4*x^4-810*I*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
))*a^2*x^2-1620*I*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*a^2*x^2
-810*I*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))^2*csgn(I/((a*x+1)^2/(-a^2*x^
2+1)+1))*a^2*x^2+810*I*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))^2*csgn(I*(a*
x+1)^2/(a^2*x^2-1))*a^2*x^2-270*I*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))*a^6*x^6+540*I*Pi*arctanh(a*x)^2*a^6*x^6-1620*I*Pi*arctanh(a*x)^2*a^4*x^4+1620*I*Pi*arctanh(a*x)^2*a^2
*x^2+270*I*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))+540*I*Pi*arcta
nh(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+270*I*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))^2-270*I*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))^2-270*I*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*a^6*x^6+540*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2
/(-a^2*x^2+1)+1))^3*a^6*x^6-270*I*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*a^6*x^6-540*I*Pi*arctanh(a
*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*a^6*x^6-270*I*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))+810*I*Pi*arctanh(a*x)^2
*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*a^4*x^4+1620*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*a^4*x^4
-810*I*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*a^2*x^2+1620*I*Pi*arctanh(
a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*a^2*x^2-810*I*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*a^
2*x^2-1620*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*a^2*x^2+360*arctanh(a*x)^3*a^6*x^6+735*arc
tanh(a*x)*a^6*x^6-540*I*Pi*arctanh(a*x)^2+270*I*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))^3-540*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3+270*I*Pi*arctanh(a*x)^2*csgn(I*(a*
x+1)^2/(a^2*x^2-1))^3+540*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2)/(a*x-1)^3/(a*x+1)^3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (183) = 366\).
time = 0.29, size = 516, normalized size = 2.41 \begin {gather*} -\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 {gather*}

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 = 0.39, size = 179, normalized size = 0.84 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 2.49, size = 493, normalized size = 2.30 \begin {gather*} {\ln \left (1-a\,x\right )}^2\,\left (\frac {5\,\ln \left (a\,x+1\right )}{128\,a}-\frac {\frac {5\,a^4\,x^5}{16}-\frac {5\,a^2\,x^3}{6}+\frac {11\,x}{16}}{4\,a^6\,x^6-12\,a^4\,x^4+12\,a^2\,x^2-4}\right )-\frac {\frac {245\,a^4\,x^5}{8}-\frac {200\,a^2\,x^3}{3}+\frac {299\,x}{8}}{144\,a^6\,x^6-432\,a^4\,x^4+432\,a^2\,x^2-144}-\ln \left (1-a\,x\right )\,\left (\frac {5\,{\ln \left (a\,x+1\right )}^2}{128\,a}+\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{192\,a^6\,x^6-576\,a^4\,x^4+576\,a^2\,x^2-192}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{192\,a^6\,x^6-576\,a^4\,x^4+576\,a^2\,x^2-192}-\frac {\ln \left (a\,x+1\right )\,\left (10\,a^4\,x^5-\frac {80\,a^2\,x^3}{3}+22\,x\right )}{64\,a^6\,x^6-192\,a^4\,x^4+192\,a^2\,x^2-64}\right )+\frac {5\,{\ln \left (a\,x+1\right )}^3}{384\,a}-\frac {5\,{\ln \left (1-a\,x\right )}^3}{384\,a}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {17}{72\,a^2}-\frac {35\,x^2}{96}+\frac {5\,a^2\,x^4}{32}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}-\frac {{\ln \left (a\,x+1\right )}^2\,\left (\frac {11\,x}{64\,a}-\frac {5\,a\,x^3}{24}+\frac {5\,a^3\,x^5}{64}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}-\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,245{}\mathrm {i}}{1152\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)^2/(a^2*x^2 - 1)^4,x)

[Out]

log(1 - a*x)^2*((5*log(a*x + 1))/(128*a) - ((11*x)/16 - (5*a^2*x^3)/6 + (5*a^4*x^5)/16)/(12*a^2*x^2 - 12*a^4*x
^4 + 4*a^6*x^6 - 4)) - ((299*x)/8 - (200*a^2*x^3)/3 + (245*a^4*x^5)/8)/(432*a^2*x^2 - 432*a^4*x^4 + 144*a^6*x^
6 - 144) - log(1 - a*x)*((5*log(a*x + 1)^2)/(128*a) + ((37*x)/2 - 35*a*x^2 + 68/(3*a) - (82*a^2*x^3)/3 + 15*a^
3*x^4 + (23*a^4*x^5)/2)/(576*a^2*x^2 - 576*a^4*x^4 + 192*a^6*x^6 - 192) - ((37*x)/2 + 35*a*x^2 - 68/(3*a) - (8
2*a^2*x^3)/3 - 15*a^3*x^4 + (23*a^4*x^5)/2)/(576*a^2*x^2 - 576*a^4*x^4 + 192*a^6*x^6 - 192) - (log(a*x + 1)*(2
2*x - (80*a^2*x^3)/3 + 10*a^4*x^5))/(192*a^2*x^2 - 192*a^4*x^4 + 64*a^6*x^6 - 64)) + (5*log(a*x + 1)^3)/(384*a
) - (5*log(1 - a*x)^3)/(384*a) - (atan(a*x*1i)*245i)/(1152*a) + (log(a*x + 1)*(17/(72*a^2) - (35*x^2)/96 + (5*
a^2*x^4)/32))/(3*a*x^2 - 1/a - 3*a^3*x^4 + a^5*x^6) - (log(a*x + 1)^2*((11*x)/(64*a) - (5*a*x^3)/24 + (5*a^3*x
^5)/64))/(3*a*x^2 - 1/a - 3*a^3*x^4 + a^5*x^6)

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