3.4.15 \(\int \frac {x^2 \tanh ^{-1}(a x)^3}{(1-a^2 x^2)^3} \, dx\) [315]

Optimal. Leaf size=215 \[ -\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {3}{128 a^3 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^4}{32 a^3} \]

[Out]

-3/128/a^3/(-a^2*x^2+1)^2+3/128/a^3/(-a^2*x^2+1)+3/32*x*arctanh(a*x)/a^2/(-a^2*x^2+1)^2-3/64*x*arctanh(a*x)/a^
2/(-a^2*x^2+1)-3/128*arctanh(a*x)^2/a^3-3/16*arctanh(a*x)^2/a^3/(-a^2*x^2+1)^2+3/16*arctanh(a*x)^2/a^3/(-a^2*x
^2+1)+1/4*x*arctanh(a*x)^3/a^2/(-a^2*x^2+1)^2-1/8*x*arctanh(a*x)^3/a^2/(-a^2*x^2+1)-1/32*arctanh(a*x)^4/a^3

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Rubi [A]
time = 0.26, antiderivative size = 215, 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, 267, 6111, 6107} \begin {gather*} -\frac {\tanh ^{-1}(a x)^4}{32 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{128 a^3}-\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 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 {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 {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} \end {gather*}

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 267

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 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 6107

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]

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)^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*}

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Mathematica [A]
time = 0.05, size = 107, normalized size = 0.50 \begin {gather*} \frac {-3 a^2 x^2+6 \left (a x+a^3 x^3\right ) \tanh ^{-1}(a x)-3 \left (1+6 a^2 x^2+a^4 x^4\right ) \tanh ^{-1}(a x)^2+16 \left (a x+a^3 x^3\right ) \tanh ^{-1}(a x)^3-4 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}{128 a^3 \left (-1+a^2 x^2\right )^2} \end {gather*}

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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 387.63, size = 2059, normalized size = 9.58

method	result	size
risch	\(-\frac {\ln \left (a x +1\right )^{4}}{512 a^{3}}+\frac {\left (x^{4} \ln \left (-a x +1\right ) a^{4}+2 a^{3} x^{3}-2 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{3}}{128 a^{3} \left (a^{2} x^{2}-1\right )^{2}}-\frac {3 \left (2 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+a^{4} x^{4}+8 a^{3} x^{3} \ln \left (-a x +1\right )-4 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+6 a^{2} x^{2}+8 a x \ln \left (-a x +1\right )+2 \ln \left (-a x +1\right )^{2}+1\right ) \ln \left (a x +1\right )^{2}}{512 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}+\frac {\left (2 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+3 x^{4} \ln \left (-a x +1\right ) a^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )^{2}-4 a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 a^{3} x^{3}+18 x^{2} \ln \left (-a x +1\right ) a^{2}+12 a \ln \left (-a x +1\right )^{2} x +2 \ln \left (-a x +1\right )^{3}+6 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{256 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {a^{4} x^{4} \ln \left (-a x +1\right )^{4}+3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+8 a^{3} x^{3} \ln \left (-a x +1\right )^{3}-2 a^{2} x^{2} \ln \left (-a x +1\right )^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )+18 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+8 a x \ln \left (-a x +1\right )^{3}+12 a^{2} x^{2}+\ln \left (-a x +1\right )^{4}+12 a x \ln \left (-a x +1\right )+3 \ln \left (-a x +1\right )^{2}}{512 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\)	\(559\)
derivativedivides	\(\text {Expression too large to display}\)	\(2059\)
default	\(\text {Expression too large to display}\)	\(2059\)

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,method=_RETURNVERBOSE)

[Out]

1/a^3*(-1/16*arctanh(a*x)^3/(a*x+1)^2+1/16*arctanh(a*x)^3/(a*x+1)-1/16*arctanh(a*x)^3*ln(a*x+1)+1/16*arctanh(a
*x)^3/(a*x-1)^2+1/16*arctanh(a*x)^3/(a*x-1)+1/16*arctanh(a*x)^3*ln(a*x-1)+1/8*arctanh(a*x)^3*ln((a*x+1)/(-a^2*
x^2+1)^(1/2))-1/1024*(3-32*I*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+32*I*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))^2-32*I*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))+32*I*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))*arctanh(a*x)^3*a^4*x^4+64*I*Pi*arctanh(a*x)^3*a^4*x^4-128*I*Pi*arctanh(a*x)^3
*a^2*x^2+18*a^2*x^2-48*a*x*arctanh(a*x)-64*I*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+3*a^4*x^4-64*I*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))*arctanh(a*x)^3*a^2*x^2-48*a^3*x^3*arctanh(a*x)+24*a^4*x^4*ar
ctanh(a*x)^2+144*a^2*x^2*arctanh(a*x)^2+32*arctanh(a*x)^4+24*arctanh(a*x)^2+32*arctanh(a*x)^4*a^4*x^4-64*arcta
nh(a*x)^4*a^2*x^2-32*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)^3*a^4*x^4+64
*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^3*a^4*x^4-32*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arct
anh(a*x)^3*a^4*x^4-64*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^3*a^4*x^4-128*I*Pi*csgn(I/((a*x+1
)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^3*a^2*x^2+64*I*Pi*arctanh(a*x)^3+64*I*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^2/
(-a^2*x^2+1)+1))^3-32*I*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))^3-32*I*Pi*a
rctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3-64*I*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2+64*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)^3*a^2*x^2+64*I*Pi*csgn(I*(a*x+1)^2
/(a^2*x^2-1))^3*arctanh(a*x)^3*a^2*x^2+128*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^3*a^2*x^2+32
*I*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))+64*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))*ar
ctanh(a*x)^3*a^2*x^2+128*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)^
3*a^2*x^2-32*I*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*arctanh(a*x)^3*a^4*x^4+32*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)^3*a^4*x^4-32*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)^3*a^4*x^4-64*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)
^3*a^4*x^4+64*I*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*arctanh(a*x)^3*a^2*x^2-64*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)^3*a^2*x^2)/(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. 657 vs. \(2 (187) = 374\).
time = 0.28, size = 657, normalized size = 3.06 \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 )^{3} - \frac {3 \, {\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 )^{2}}{64 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} + \frac {1}{512} \, {\left (\frac {{\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{4} - 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{4} - 12 \, a^{2} x^{2} + 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 )^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 2 \, {\left (2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )\right )} a^{2}}{a^{10} x^{4} - 2 \, a^{8} x^{2} + a^{6}} + \frac {4 \, {\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 \operatorname {artanh}\left (a x\right )}{a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}}\right )} a \end {gather*}

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 = 0.42, size = 161, normalized size = 0.75 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 {gather*}

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 = 0.00, 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)^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)

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Mupad [B]
time = 3.19, size = 831, normalized size = 3.87 \begin {gather*} \frac {3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )}-\frac {3\,{\ln \left (1-a\,x\right )}^2}{512\,a^3}-\frac {{\ln \left (a\,x+1\right )}^4}{512\,a^3}-\frac {{\ln \left (1-a\,x\right )}^4}{512\,a^3}-\frac {3\,x^2}{2\,\left (64\,a^5\,x^4-128\,a^3\,x^2+64\,a\right )}-\frac {x\,{\ln \left (1-a\,x\right )}^3}{8\,\left (8\,a^6\,x^4-16\,a^4\,x^2+8\,a^2\right )}-\frac {6\,x^2\,{\ln \left (1-a\,x\right )}^2}{128\,a^5\,x^4-256\,a^3\,x^2+128\,a}-\frac {3\,{\ln \left (a\,x+1\right )}^2}{512\,a^3}+\frac {x^3\,{\ln \left (a\,x+1\right )}^3}{64\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}-\frac {x^3\,{\ln \left (1-a\,x\right )}^3}{8\,\left (8\,a^4\,x^4-16\,a^2\,x^2+8\right )}+\frac {3\,x\,\ln \left (a\,x+1\right )}{128\,\left (a^6\,x^4-2\,a^4\,x^2+a^2\right )}+\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^3}{128\,a^3}+\frac {{\ln \left (a\,x+1\right )}^3\,\ln \left (1-a\,x\right )}{128\,a^3}-\frac {3\,x\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {3\,x^2\,{\ln \left (a\,x+1\right )}^2}{64\,\left (a^5\,x^4-2\,a^3\,x^2+a\right )}+\frac {x\,{\ln \left (a\,x+1\right )}^3}{64\,\left (a^6\,x^4-2\,a^4\,x^2+a^2\right )}-\frac {3\,{\ln \left (a\,x+1\right )}^2\,{\ln \left (1-a\,x\right )}^2}{256\,a^3}+\frac {3\,x^3\,\ln \left (a\,x+1\right )}{128\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}-\frac {3\,a\,x^3\,\ln \left (1-a\,x\right )}{128\,a^5\,x^4-256\,a^3\,x^2+128\,a}+\frac {6\,x\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {6\,x\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}+\frac {6\,x^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{64\,a^5\,x^4-128\,a^3\,x^2+64\,a}+\frac {6\,a^2\,x^3\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {6\,a^2\,x^3\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {3\,a^2\,x^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{2\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )}+\frac {3\,a^4\,x^4\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*log(a*x + 1)*log(1 - a*x))/(4*(64*a^3 - 128*a^5*x^2 + 64*a^7*x^4)) - (3*log(1 - a*x)^2)/(512*a^3) - log(a*x
+ 1)^4/(512*a^3) - log(1 - a*x)^4/(512*a^3) - (3*x^2)/(2*(64*a - 128*a^3*x^2 + 64*a^5*x^4)) - (x*log(1 - a*x)
^3)/(8*(8*a^2 - 16*a^4*x^2 + 8*a^6*x^4)) - (6*x^2*log(1 - a*x)^2)/(128*a - 256*a^3*x^2 + 128*a^5*x^4) - (3*log
(a*x + 1)^2)/(512*a^3) + (x^3*log(a*x + 1)^3)/(64*(a^4*x^4 - 2*a^2*x^2 + 1)) - (x^3*log(1 - a*x)^3)/(8*(8*a^4*
x^4 - 16*a^2*x^2 + 8)) + (3*x*log(a*x + 1))/(128*(a^2 - 2*a^4*x^2 + a^6*x^4)) + (log(a*x + 1)*log(1 - a*x)^3)/
(128*a^3) + (log(a*x + 1)^3*log(1 - a*x))/(128*a^3) - (3*x*log(1 - a*x))/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4)
- (3*x^2*log(a*x + 1)^2)/(64*(a - 2*a^3*x^2 + a^5*x^4)) + (x*log(a*x + 1)^3)/(64*(a^2 - 2*a^4*x^2 + a^6*x^4))
- (3*log(a*x + 1)^2*log(1 - a*x)^2)/(256*a^3) + (3*x^3*log(a*x + 1))/(128*(a^4*x^4 - 2*a^2*x^2 + 1)) - (3*a*x
^3*log(1 - a*x))/(128*a - 256*a^3*x^2 + 128*a^5*x^4) + (6*x*log(a*x + 1)*log(1 - a*x)^2)/(128*a^2 - 256*a^4*x^
2 + 128*a^6*x^4) - (6*x*log(a*x + 1)^2*log(1 - a*x))/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4) + (6*x^2*log(a*x +
1)*log(1 - a*x))/(64*a - 128*a^3*x^2 + 64*a^5*x^4) + (6*a^2*x^3*log(a*x + 1)*log(1 - a*x)^2)/(128*a^2 - 256*a^
4*x^2 + 128*a^6*x^4) - (6*a^2*x^3*log(a*x + 1)^2*log(1 - a*x))/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4) - (3*a^2*
x^2*log(a*x + 1)*log(1 - a*x))/(2*(64*a^3 - 128*a^5*x^2 + 64*a^7*x^4)) + (3*a^4*x^4*log(a*x + 1)*log(1 - a*x))
/(4*(64*a^3 - 128*a^5*x^2 + 64*a^7*x^4))

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