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

Optimal. Leaf size=188 \[ -\frac{45 x}{256 a \left (1-a^2 x^2\right )}-\frac{3 x}{128 a \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac{9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \tanh ^{-1}(a x)^3}{32 a^2}-\frac{45 \tanh ^{-1}(a x)}{256 a^2} \]

[Out]

(-3*x)/(128*a*(1 - a^2*x^2)^2) - (45*x)/(256*a*(1 - a^2*x^2)) - (45*ArcTanh[a*x])/(256*a^2) + (3*ArcTanh[a*x])
/(32*a^2*(1 - a^2*x^2)^2) + (9*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)) - (3*x*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)
^2) - (9*x*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^3)/(32*a^2) + ArcTanh[a*x]^3/(4*a^2*(1 - a^2
*x^2)^2)

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Rubi [A]  time = 0.163647, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5994, 5964, 5956, 199, 206} \[ -\frac{45 x}{256 a \left (1-a^2 x^2\right )}-\frac{3 x}{128 a \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac{9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \tanh ^{-1}(a x)^3}{32 a^2}-\frac{45 \tanh ^{-1}(a x)}{256 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(128*a*(1 - a^2*x^2)^2) - (45*x)/(256*a*(1 - a^2*x^2)) - (45*ArcTanh[a*x])/(256*a^2) + (3*ArcTanh[a*x])
/(32*a^2*(1 - a^2*x^2)^2) + (9*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)) - (3*x*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)
^2) - (9*x*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^3)/(32*a^2) + ArcTanh[a*x]^3/(4*a^2*(1 - a^2
*x^2)^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 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 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{x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx &=\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx}{4 a}\\ &=\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \int \frac{1}{\left (1-a^2 x^2\right )^3} \, dx}{32 a}-\frac{9 \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{16 a}\\ &=-\frac{3 x}{128 a \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac{9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac{9}{16} \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac{9 \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx}{128 a}\\ &=-\frac{3 x}{128 a \left (1-a^2 x^2\right )^2}-\frac{9 x}{256 a \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac{9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac{9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{9 \int \frac{1}{1-a^2 x^2} \, dx}{256 a}-\frac{9 \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx}{32 a}\\ &=-\frac{3 x}{128 a \left (1-a^2 x^2\right )^2}-\frac{45 x}{256 a \left (1-a^2 x^2\right )}-\frac{9 \tanh ^{-1}(a x)}{256 a^2}+\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac{9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac{9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{9 \int \frac{1}{1-a^2 x^2} \, dx}{64 a}\\ &=-\frac{3 x}{128 a \left (1-a^2 x^2\right )^2}-\frac{45 x}{256 a \left (1-a^2 x^2\right )}-\frac{45 \tanh ^{-1}(a x)}{256 a^2}+\frac{3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac{9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac{9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac{\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0864559, size = 148, normalized size = 0.79 \[ \frac{90 a^3 x^3-45 a^4 x^4 \log (a x+1)+90 a^2 x^2 \log (a x+1)+45 \left (a^2 x^2-1\right )^2 \log (1-a x)+48 a x \left (3 a^2 x^2-5\right ) \tanh ^{-1}(a x)^2+\left (-48 a^4 x^4+96 a^2 x^2+80\right ) \tanh ^{-1}(a x)^3-48 \left (3 a^2 x^2-4\right ) \tanh ^{-1}(a x)-102 a x-45 \log (a x+1)}{512 a^2 \left (a^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-102*a*x + 90*a^3*x^3 - 48*(-4 + 3*a^2*x^2)*ArcTanh[a*x] + 48*a*x*(-5 + 3*a^2*x^2)*ArcTanh[a*x]^2 + (80 + 96*
a^2*x^2 - 48*a^4*x^4)*ArcTanh[a*x]^3 + 45*(-1 + a^2*x^2)^2*Log[1 - a*x] - 45*Log[1 + a*x] + 90*a^2*x^2*Log[1 +
a*x] - 45*a^4*x^4*Log[1 + a*x])/(512*a^2*(-1 + a^2*x^2)^2)

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Maple [C]  time = 0.348, size = 2582, normalized size = 13.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a^2/(a^2*x^2-1)^2*arctanh(a*x)^3-3/64/a^2*arctanh(a*x)^2/(a*x-1)^2+9/64/a^2*arctanh(a*x)^2/(a*x-1)+9/64/a^
2*arctanh(a*x)^2*ln(a*x-1)+3/64/a^2*arctanh(a*x)^2/(a*x+1)^2+9/64/a^2*arctanh(a*x)^2/(a*x+1)-9/64/a^2*arctanh(
a*x)^2*ln(a*x+1)+9/32/a^2*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-9/64*I/a^2/(a*x-1)^2/(a*x+1)^2*Pi*arct
anh(a*x)^2*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3+9/128*I/a^2/(a*x-1)^2/(a*x+1)^2*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))^2-9/64*I/(a*x-1)^2/(a*x+1)^2*a
rctanh(a*x)^2*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))*x^2-9/32*I/(a*x-1)^2/(a*x+
1)^2*arctanh(a*x)^2*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*x^2+9/64*I/(a*x-1)^2
/(a*x+1)^2*arctanh(a*x)^2*Pi*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*x^2-9/64*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*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*x^2+9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*csgn(I*(a*x+1)^
2/(a^2*x^2-1))^3*x^4+9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2
/(-a^2*x^2+1)+1))^3*x^4-9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*
x^4+9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*x^4+9/128*I/a^2/(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))+9/64*I/a^
2/(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))^2-9/1
28*I/a^2/(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)/((a*
x+1)^2/(-a^2*x^2+1)+1))^2-9/128*I/a^2/(a*x-1)^2/(a*x+1)^2*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))+9/128*I*a^2/(a*x-1)^2/
(a*x+1)^2*arctanh(a*x)^2*Pi*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*x^4+9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*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))*x^4+9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*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*x^4-9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*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*x^4+9/64*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^
2*Pi*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))*x^2-9/64*I/a^2/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2+9/32*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^
2*Pi*x^2-9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*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))*x^4-3/32*a^2/(a*x-1)^2/(a*x+1)^2*arctan
h(a*x)^3*x^4-45/256*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)*x^4+9/64*I/a^2/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*
csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2-9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*x^4+45/256*a/(a*x-1)^2/(
a*x+1)^2*x^3-51/256/a/(a*x-1)^2/(a*x+1)^2*x-3/32/a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3+51/256/a^2/(a*x-1)^2/(
a*x+1)^2*arctanh(a*x)+3/16/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*x^2+9/128/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)*x^2-9
/64*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*x^2-9/64*I/(a*x-1)^2/(a*x+1)^2*arc
tanh(a*x)^2*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*x^2+9/32*I/(a*x-1)^2/(a*x+1)^2*arcta
nh(a*x)^2*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*x^2-9/32*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*csgn(I/((a*
x+1)^2/(-a^2*x^2+1)+1))^2*x^2+9/128*I/a^2/(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+9/128*I/a^2/(a*x-1)^2/(a*x+1)^2*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

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Maxima [B]  time = 1.01383, size = 570, normalized size = 3.03 \begin{align*} \frac{3 \,{\left (\frac{2 \,{\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac{3 \, \log \left (a x + 1\right )}{a} + \frac{3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname{artanh}\left (a x\right )^{2}}{64 \, a} + \frac{3 \,{\left (\frac{{\left (30 \, 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} - 34 \, a x - 3 \,{\left (5 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 2 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 5\right )} \log \left (a x + 1\right ) + 15 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}} - \frac{4 \,{\left (12 \, a^{2} x^{2} - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 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 )^{2} - 16\right )} a \operatorname{artanh}\left (a x\right )}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}}\right )}}{512 \, a} + \frac{\operatorname{artanh}\left (a x\right )^{3}}{4 \,{\left (a^{2} x^{2} - 1\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/64*(2*(3*a^2*x^3 - 5*x)/(a^4*x^4 - 2*a^2*x^2 + 1) - 3*log(a*x + 1)/a + 3*log(a*x - 1)/a)*arctanh(a*x)^2/a +
3/512*((30*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*l
og(a*x - 1) + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 - 34*a*x - 3*(5*a^4*x^4 - 10*a^2*x^2 + 2*(a^4*x^4 - 2
*a^2*x^2 + 1)*log(a*x - 1)^2 + 5)*log(a*x + 1) + 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*a^2/(a^7*x^4 - 2*a
^5*x^2 + a^3) - 4*(12*a^2*x^2 - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a
*x + 1)*log(a*x - 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 16)*a*arctanh(a*x)/(a^6*x^4 - 2*a^4*x^2 +
a^2))/a + 1/4*arctanh(a*x)^3/((a^2*x^2 - 1)^2*a^2)

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Fricas [A]  time = 2.05961, size = 312, normalized size = 1.66 \begin{align*} \frac{90 \, a^{3} x^{3} - 2 \,{\left (3 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 5\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 12 \,{\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 102 \, a x - 3 \,{\left (15 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 17\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{512 \,{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512*(90*a^3*x^3 - 2*(3*a^4*x^4 - 6*a^2*x^2 - 5)*log(-(a*x + 1)/(a*x - 1))^3 + 12*(3*a^3*x^3 - 5*a*x)*log(-(a
*x + 1)/(a*x - 1))^2 - 102*a*x - 3*(15*a^4*x^4 - 6*a^2*x^2 - 17)*log(-(a*x + 1)/(a*x - 1)))/(a^6*x^4 - 2*a^4*x
^2 + a^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x*atanh(a*x)**3/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x \operatorname{artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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