3.315 \(\int \frac{x^2 \tanh ^{-1}(a x)^3}{(1-a^2 x^2)^3} \, dx\)
Optimal. Leaf size=215 \[ \frac{3}{128 a^3 \left (1-a^2 x^2\right )}-\frac{3}{128 a^3 \left (1-a^2 x^2\right )^2}-\frac{x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}-\frac{3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}+\frac{3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)^4}{32 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{128 a^3} \]
[Out]
-3/(128*a^3*(1 - a^2*x^2)^2) + 3/(128*a^3*(1 - a^2*x^2)) + (3*x*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)^2) - (3*x*
ArcTanh[a*x])/(64*a^2*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^2)/(128*a^3) - (3*ArcTanh[a*x]^2)/(16*a^3*(1 - a^2*x^2)
^2) + (3*ArcTanh[a*x]^2)/(16*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(4*a^2*(1 - a^2*x^2)^2) - (x*ArcTanh[a*x]
^3)/(8*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^4/(32*a^3)
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Rubi [A] time = 0.358817, antiderivative size = 215, normalized size of antiderivative = 1.,
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, 261, 5964, 5960} \[ \frac{3}{128 a^3 \left (1-a^2 x^2\right )}-\frac{3}{128 a^3 \left (1-a^2 x^2\right )^2}-\frac{x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}-\frac{3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}+\frac{3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)^4}{32 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{128 a^3} \]
Antiderivative was successfully verified.
[In]
Int[(x^2*ArcTanh[a*x]^3)/(1 - a^2*x^2)^3,x]
[Out]
-3/(128*a^3*(1 - a^2*x^2)^2) + 3/(128*a^3*(1 - a^2*x^2)) + (3*x*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)^2) - (3*x*
ArcTanh[a*x])/(64*a^2*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^2)/(128*a^3) - (3*ArcTanh[a*x]^2)/(16*a^3*(1 - a^2*x^2)
^2) + (3*ArcTanh[a*x]^2)/(16*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(4*a^2*(1 - a^2*x^2)^2) - (x*ArcTanh[a*x]
^3)/(8*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^4/(32*a^3)
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]
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 261
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 5960
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 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]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
Rubi steps
\begin{align*} \int \frac{x^2 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx &=\frac{\int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx}{a^2}-\frac{\int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}\\ &=-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{8 a^3}+\frac{3 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx}{8 a^2}+\frac{3 \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx}{4 a^2}+\frac{3 \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{2 a}\\ &=-\frac{3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{32 a^3}+\frac{9 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{32 a^2}-\frac{3 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}-\frac{9 \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{8 a}\\ &=-\frac{3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{39 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac{39 \tanh ^{-1}(a x)^2}{128 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{32 a^3}+\frac{9 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{8 a^2}-\frac{9 \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx}{64 a}+\frac{3 \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx}{4 a}\\ &=-\frac{3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac{39}{128 a^3 \left (1-a^2 x^2\right )}+\frac{3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{32 a^3}-\frac{9 \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx}{16 a}\\ &=-\frac{3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac{3}{128 a^3 \left (1-a^2 x^2\right )}+\frac{3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^4}{32 a^3}\\ \end{align*}
Mathematica [A] time = 0.122521, size = 107, normalized size = 0.5 \[ \frac{-3 a^2 x^2-4 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4+16 \left (a^3 x^3+a x\right ) \tanh ^{-1}(a x)^3-3 \left (a^4 x^4+6 a^2 x^2+1\right ) \tanh ^{-1}(a x)^2+6 \left (a^3 x^3+a x\right ) \tanh ^{-1}(a x)}{128 a^3 \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^2*ArcTanh[a*x]^3)/(1 - a^2*x^2)^3,x]
[Out]
(-3*a^2*x^2 + 6*(a*x + a^3*x^3)*ArcTanh[a*x] - 3*(1 + 6*a^2*x^2 + a^4*x^4)*ArcTanh[a*x]^2 + 16*(a*x + a^3*x^3)
*ArcTanh[a*x]^3 - 4*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4)/(128*a^3*(-1 + a^2*x^2)^2)
________________________________________________________________________________________
Maple [C] time = 0.432, size = 2646, normalized size = 12.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*arctanh(a*x)^3/(-a^2*x^2+1)^3,x)
[Out]
1/16/a^3*arctanh(a*x)^3*ln(a*x-1)-1/16/a^3*arctanh(a*x)^3/(a*x+1)^2+1/16/a^3*arctanh(a*x)^3/(a*x+1)-1/16/a^3*a
rctanh(a*x)^3*ln(a*x+1)+1/8/a^3*arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-3/1024/a^3/(a*x-1)^2/(a*x+1)^2+1
/16/a^3*arctanh(a*x)^3/(a*x-1)^2+1/16/a^3*arctanh(a*x)^3/(a*x-1)-1/16*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*P
i*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*x^4+1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*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+1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^
2*x^2-1))^3*x^4+1/16*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*x^4+1/8*I/
a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*x^2-1/16*I/a/(a*x-1)^2/(a*x+1)^2*
arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*x^2-1/16*I/a/(a*x-1)^2/(a*x+1)^2*
arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*x^2-1/8*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I/((a
*x+1)^2/(-a^2*x^2+1)+1))^2*x^2+1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*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))^2-1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)
^3*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+1/16*I/a^3/(a*x-1)
^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+1/32*I/a^3/(
a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*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*a
/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^4*x^4-3/128*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*x^4+1/16/a/(a*x-1)^2/(a*x+1
)^2*arctanh(a*x)^4*x^2-9/64/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*x^2+3/64/a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)
*x+1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*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^4-1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*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))*x^4+1/16*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^
3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*x^4+1/32*I*a/(a*x-1)^2/(a*x+1)^2*arcta
nh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*x^4-1/16*I/a/(a*x-1)^2/(a*x+1)
^2*arctanh(a*x)^3*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+1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*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))*x^2-1/8*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2
-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*x^2-1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/
(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*x^2-1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*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))-1/32*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*x^4+1/16*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^
3*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))*csgn(I*(a*x+1
)^2/(a^2*x^2-1))*x^2+1/8*I/a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*x^2-1/16*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arcta
nh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3+1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)
^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3+1/32*I/a^3/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^3*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)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2-3/1024*a
/(a*x-1)^2/(a*x+1)^2*x^4-9/512/a/(a*x-1)^2/(a*x+1)^2*x^2+3/64/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)*x^3-1/32/a^3/(a
*x-1)^2/(a*x+1)^2*arctanh(a*x)^4-3/128/a^3/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2-1/16*I/a^3/(a*x-1)^2/(a*x+1)^2*P
i*arctanh(a*x)^3-1/16*I*a/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*Pi*x^4
________________________________________________________________________________________
Maxima [B] time = 1.06039, size = 887, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arctanh(a*x)^3/(-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)^3 - 3/64
*(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)^2/(a^8*x^4 - 2*a^6*x^2 + a^4) + 1/512*(((a^4*x^4 -
2*a^2*x^2 + 1)*log(a*x + 1)^4 - 4*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3*log(a*x - 1) + (a^4*x^4 - 2*a^2*x^
2 + 1)*log(a*x - 1)^4 - 12*a^2*x^2 + 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)^2 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 2*(2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 +
3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + 1))*a^2/(a^10*x^4 - 2*a^8*x^2 + a^6) + 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*arctanh(a*x)/(a^9*x^4 - 2*a^7*x^2 + a^5))*a
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Fricas [A] time = 2.03667, size = 348, normalized size = 1.62 \begin{align*} -\frac{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{4} + 12 \, a^{2} x^{2} - 8 \,{\left (a^{3} x^{3} + a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 3 \,{\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 12 \,{\left (a^{3} x^{3} + a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{512 \,{\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arctanh(a*x)^3/(-a^2*x^2+1)^3,x, algorithm="fricas")
[Out]
-1/512*((a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^4 + 12*a^2*x^2 - 8*(a^3*x^3 + a*x)*log(-(a*x + 1)/
(a*x - 1))^3 + 3*(a^4*x^4 + 6*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 - 12*(a^3*x^3 + a*x)*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., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2} \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**2*atanh(a*x)**3/(-a**2*x**2+1)**3,x)
[Out]
-Integral(x**2*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^{2} \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^2*arctanh(a*x)^3/(-a^2*x^2+1)^3,x, algorithm="giac")
[Out]
integrate(-x^2*arctanh(a*x)^3/(a^2*x^2 - 1)^3, x)