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

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

[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 = 0.25, 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 = {6028, 5956, 5994, 199, 206, 5964} \[ -\frac {x}{64 a^2 \left (1-a^2 x^2\right )}+\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}-\frac {\tanh ^{-1}(a x)}{64 a^3} \]

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 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])

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 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 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 6028

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 = 0.12, size = 121, normalized size = 0.74 \[ \frac {48 \left (a^3 x^3+a x\right ) \tanh ^{-1}(a x)^2+6 a x \left (a^2 x^2+1\right )+3 \left (a^2 x^2-1\right )^2 \log (1-a x)-3 \left (a^2 x^2-1\right )^2 \log (a x+1)-16 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^3-48 a^2 x^2 \tanh ^{-1}(a x)}{384 a^3 \left (a^2 x^2-1\right )^2} \]

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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fricas [A]  time = 0.62, size = 136, normalized size = 0.83 \[ \frac {6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, a x - 3 \, {\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{384 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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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maple [C]  time = 0.89, size = 2571, normalized size = 15.77 \[ \text {Expression too large to display} \]

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)

[Out]

-1/16*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2+1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*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))+1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(
I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2-1/32*I/a^3/(a*x-1)^2/
(a*x+1)^2*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)/(1+(a*x+1)^2/(-a^2*x^2+
1)))^2+1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*
Pi*x^4+1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^4+1/16*I*a/(a*x-1)^2/(
a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi*x^4+1/8*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2
*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi*x^2-1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2
*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi*x^2-1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2
*x^2-1))^3*Pi*x^2-1/8*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi*x^2-1/16*
I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi*x^4+1/16*I/a^3/(a*x-1)^2/(a*x+1
)^2*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+1/64/(a*x-1)^2/(a*x+1
)^2*x^3+1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x
^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi*x^2-1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*
x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1
)^2/(a^2*x^2-1))*Pi*x^4+1/8/a^3*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+1/16/a^3*arctanh(a*x)^2/(a*x-1)^
2+1/16/a^3*arctanh(a*x)^2/(a*x-1)+1/16/a^3*arctanh(a*x)^2*ln(a*x-1)-1/16/a^3*arctanh(a*x)^2/(a*x+1)^2+1/16/a^3
*arctanh(a*x)^2/(a*x+1)-1/16/a^3*arctanh(a*x)^2*ln(a*x+1)+1/64/a^2/(a*x-1)^2/(a*x+1)^2*x-1/24/a^3/(a*x-1)^2/(a
*x+1)^2*arctanh(a*x)^3-1/64/a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)+1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*c
sgn(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^4-1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(
a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi*x^2+1/
16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*csgn(I*(a
*x+1)^2/(a^2*x^2-1))*Pi*x^2-1/8*I/a/(a*x-1)^2/(a*x+1)^2*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^2-1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*c
sgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*Pi*x^2-1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^
2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))+1/32*I
*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+
1)^2/(-a^2*x^2+1)))^2*Pi*x^4-1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+
1)^2/(-a^2*x^2+1)))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi*x^4+1/16*I*a/(a*x-1)^2/(a*x+1)^2*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^4-1/16*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2
*Pi*x^4+1/8*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*x^2+1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csg
n(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3+1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(
I*(a*x+1)^2/(a^2*x^2-1))^3+1/16*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))
^2-1/16*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3-1/24*a/(a*x-1)^2/(a*x
+1)^2*arctanh(a*x)^3*x^4-1/64*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)*x^4+1/12/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3
*x^2-3/32/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)*x^2

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maxima [B]  time = 0.33, size = 388, normalized size = 2.38 \[ \frac {1}{16} \, {\left (\frac {2 \, {\left (a^{2} x^{3} + x\right )}}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left (6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 6 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{384 \, {\left (a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}\right )}} - \frac {{\left (4 \, a^{2} x^{2} - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2}\right )} a \operatorname {artanh}\left (a x\right )}{32 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]

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

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

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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