3.4.9 \(\int \frac {x^2 \tanh ^{-1}(a x)^2}{(1-a^2 x^2)^3} \, dx\) [309]

Optimal. Leaf size=163 \[ \frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{64 a^3}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3} \]

[Out]

1/32*x/a^2/(-a^2*x^2+1)^2-1/64*x/a^2/(-a^2*x^2+1)-1/64*arctanh(a*x)/a^3-1/8*arctanh(a*x)/a^3/(-a^2*x^2+1)^2+1/
8*arctanh(a*x)/a^3/(-a^2*x^2+1)+1/4*x*arctanh(a*x)^2/a^2/(-a^2*x^2+1)^2-1/8*x*arctanh(a*x)^2/a^2/(-a^2*x^2+1)-
1/24*arctanh(a*x)^3/a^3

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Rubi [A]
time = 0.18, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6175, 6103, 6141, 205, 212, 6111} \begin {gather*} -\frac {\tanh ^{-1}(a x)^3}{24 a^3}-\frac {\tanh ^{-1}(a x)}{64 a^3}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}+\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(32*a^2*(1 - a^2*x^2)^2) - x/(64*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]/(64*a^3) - ArcTanh[a*x]/(8*a^3*(1 - a^2*x
^2)^2) + ArcTanh[a*x]/(8*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(4*a^2*(1 - a^2*x^2)^2) - (x*ArcTanh[a*x]^2)/
(8*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^3/(24*a^3)

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]

Rule 6175

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

Rubi steps

\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {\int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx}{a^2}-\frac {\int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{6 a^3}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3} \, dx}{8 a^2}+\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{4 a^2}+\frac {\int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{a}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}+\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{32 a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}-\frac {3 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{4 a}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {13 x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}+\frac {3 \int \frac {1}{1-a^2 x^2} \, dx}{64 a^2}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a^2}+\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{8 a^2}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {13 \tanh ^{-1}(a x)}{64 a^3}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}+\frac {3 \int \frac {1}{1-a^2 x^2} \, dx}{16 a^2}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{64 a^3}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 121, normalized size = 0.74 \begin {gather*} \frac {6 a x \left (1+a^2 x^2\right )-48 a^2 x^2 \tanh ^{-1}(a x)+48 \left (a x+a^3 x^3\right ) \tanh ^{-1}(a x)^2-16 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+3 \left (-1+a^2 x^2\right )^2 \log (1-a x)-3 \left (-1+a^2 x^2\right )^2 \log (1+a x)}{384 a^3 \left (-1+a^2 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*x*(1 + a^2*x^2) - 48*a^2*x^2*ArcTanh[a*x] + 48*(a*x + a^3*x^3)*ArcTanh[a*x]^2 - 16*(-1 + a^2*x^2)^2*ArcTa
nh[a*x]^3 + 3*(-1 + a^2*x^2)^2*Log[1 - a*x] - 3*(-1 + a^2*x^2)^2*Log[1 + a*x])/(384*a^3*(-1 + a^2*x^2)^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 237.51, size = 2028, normalized size = 12.44

method	result	size
risch	\(-\frac {\ln \left (a x +1\right )^{3}}{192 a^{3}}+\frac {\left (x^{4} \ln \left (-a x +1\right ) a^{4}+2 a^{3} x^{3}-2 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{64 a^{3} \left (a^{2} x^{2}-1\right )^{2}}-\frac {\left (a^{4} x^{4} \ln \left (-a x +1\right )^{2}+4 a^{3} x^{3} \ln \left (-a x +1\right )-2 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a^{2} x^{2}+4 a x \ln \left (-a x +1\right )+\ln \left (-a x +1\right )^{2}\right ) \ln \left (a x +1\right )}{64 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {-2 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+3 \ln \left (a x +1\right ) a^{4} x^{4}-3 x^{4} \ln \left (-a x +1\right ) a^{4}-12 a^{3} x^{3} \ln \left (-a x +1\right )^{2}+4 a^{2} x^{2} \ln \left (-a x +1\right )^{3}-6 a^{3} x^{3}-6 a^{2} x^{2} \ln \left (a x +1\right )-18 x^{2} \ln \left (-a x +1\right ) a^{2}-12 a \ln \left (-a x +1\right )^{2} x -2 \ln \left (-a x +1\right )^{3}-6 a x +3 \ln \left (a x +1\right )-3 \ln \left (-a x +1\right )}{384 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\)	\(394\)
derivativedivides	\(\text {Expression too large to display}\)	\(2028\)
default	\(\text {Expression too large to display}\)	\(2028\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(-1/16*arctanh(a*x)^2/(a*x+1)^2+1/16*arctanh(a*x)^2/(a*x+1)-1/16*arctanh(a*x)^2*ln(a*x+1)+1/16*arctanh(a
*x)^2/(a*x-1)^2+1/16*arctanh(a*x)^2/(a*x-1)+1/16*arctanh(a*x)^2*ln(a*x-1)+1/8*arctanh(a*x)^2*ln((a*x+1)/(-a^2*
x^2+1)^(1/2))+1/192*(16*arctanh(a*x)^3*a^2*x^2-24*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2*a^2
*x^2+6*I*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3-12*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+
1)+1))^3+12*I*Pi*arctanh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2+6*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+3*a*x+12*I*Pi*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))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)^2*a^2*x^2+6*I*Pi*csgn(I*(a*x+1)
^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^2*a^4*x^4-6*I*Pi*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))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)^2*a^4*x^4-
3*a^4*x^4*arctanh(a*x)-8*arctanh(a*x)^3-18*a^2*x^2*arctanh(a*x)-3*arctanh(a*x)+24*I*Pi*arctanh(a*x)^2*a^2*x^2-
6*I*Pi*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
+6*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))+12*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+6*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+3*a^3*x^3-8*arctanh(a*x)^3*a^4*x^4
-12*I*Pi*arctanh(a*x)^2*a^4*x^4-12*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arc
tanh(a*x)^2*a^2*x^2-24*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^2*
a^2*x^2+6*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^2*a^4*x^4+12*I*
Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^2*a^4*x^4+12*I*Pi*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))*arctanh(a*x)^2*a^2*x^2-12*I*Pi
*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))*arctanh(a*x)^2*
a^2*x^2-12*I*Pi*arctanh(a*x)^2-6*I*Pi*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))*arctanh(a*x)^2*a^4*x^4+6*I*Pi*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))*arctanh(a*x)^2*a^4*x^4+6*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2*a^
4*x^4-12*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^2*a^4*x^4+12*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+
1)+1))^2*arctanh(a*x)^2*a^4*x^4-12*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x
)^2*a^2*x^2-12*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2*a^2*x^2+24*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2
+1)+1))^3*arctanh(a*x)^2*a^2*x^2-6*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)))/(a*x+1)^2/(a*x-1)^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (141) = 282\).
time = 0.29, size = 388, normalized size = 2.38 \begin {gather*} \frac {1}{16} \, {\left (\frac {2 \, {\left (a^{2} x^{3} + x\right )}}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left (6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 6 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{384 \, {\left (a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}\right )}} - \frac {{\left (4 \, a^{2} x^{2} - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2}\right )} a \operatorname {artanh}\left (a x\right )}{32 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*(a^2*x^3 + x)/(a^6*x^4 - 2*a^4*x^2 + a^2) - log(a*x + 1)/a^3 + log(a*x - 1)/a^3)*arctanh(a*x)^2 + 1/38
4*(6*a^3*x^3 - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2*log(a*x
- 1) + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 + 6*a*x - 3*(a^4*x^4 - 2*a^2*x^2 + 2*(a^4*x^4 - 2*a^2*x^2 +
1)*log(a*x - 1)^2 + 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*a^2/(a^9*x^4 - 2*a^7*x^2 + a^
5) - 1/32*(4*a^2*x^2 - (a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log
(a*x - 1) - (a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2)*a*arctanh(a*x)/(a^8*x^4 - 2*a^6*x^2 + a^4)

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Fricas [A]
time = 0.42, size = 136, normalized size = 0.83 \begin {gather*} \frac {6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, a x - 3 \, {\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{384 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(6*a^3*x^3 - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^3 + 12*(a^3*x^3 + a*x)*log(-(a*x + 1)
/(a*x - 1))^2 + 6*a*x - 3*(a^4*x^4 + 6*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)))/(a^7*x^4 - 2*a^5*x^2 + a^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x**2*atanh(a*x)**2/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), 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(x^2*arctanh(a*x)^2/(-a^2*x^2+1)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.97, size = 350, normalized size = 2.15 \begin {gather*} \ln \left (1-a\,x\right )\,\left (\frac {\frac {3\,a\,x^3}{2}-\frac {x}{2\,a}+x^2}{32\,a^5\,x^4-64\,a^3\,x^2+32\,a}+\frac {\frac {x}{2\,a}-\frac {3\,a\,x^3}{2}+x^2}{32\,a^5\,x^4-64\,a^3\,x^2+32\,a}+\frac {{\ln \left (a\,x+1\right )}^2}{64\,a^3}-\frac {\ln \left (a\,x+1\right )\,\left (2\,a^2\,x^3+2\,x\right )}{32\,a^6\,x^4-64\,a^4\,x^2+32\,a^2}\right )+\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{8\,a^4\,x^4-16\,a^2\,x^2+8}-{\ln \left (1-a\,x\right )}^2\,\left (\frac {\ln \left (a\,x+1\right )}{64\,a^3}-\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{4\,a^4\,x^4-8\,a^2\,x^2+4}\right )-\frac {{\ln \left (a\,x+1\right )}^3}{192\,a^3}+\frac {{\ln \left (1-a\,x\right )}^3}{192\,a^3}+\frac {{\ln \left (a\,x+1\right )}^2\,\left (\frac {x}{32\,a^3}+\frac {x^3}{32\,a}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4}-\frac {x^2\,\ln \left (a\,x+1\right )}{16\,a^2\,\left (\frac {1}{a}-2\,a\,x^2+a^3\,x^4\right )}+\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(1 - a*x)*(((3*a*x^3)/2 - x/(2*a) + x^2)/(32*a - 64*a^3*x^2 + 32*a^5*x^4) + (x/(2*a) - (3*a*x^3)/2 + x^2)/(
32*a - 64*a^3*x^2 + 32*a^5*x^4) + log(a*x + 1)^2/(64*a^3) - (log(a*x + 1)*(2*x + 2*a^2*x^3))/(32*a^2 - 64*a^4*
x^2 + 32*a^6*x^4)) + (x/(8*a^2) + x^3/8)/(8*a^4*x^4 - 16*a^2*x^2 + 8) - log(1 - a*x)^2*(log(a*x + 1)/(64*a^3)
- (x/(8*a^2) + x^3/8)/(4*a^4*x^4 - 8*a^2*x^2 + 4)) - log(a*x + 1)^3/(192*a^3) + log(1 - a*x)^3/(192*a^3) + (at
an(a*x*1i)*1i)/(64*a^3) + (log(a*x + 1)^2*(x/(32*a^3) + x^3/(32*a)))/(1/a - 2*a*x^2 + a^3*x^4) - (x^2*log(a*x
+ 1))/(16*a^2*(1/a - 2*a*x^2 + a^3*x^4))

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