3.347 \(\int \frac{\tanh ^{-1}(a x)^3}{(1-a^2 x^2)^4} \, dx\)
Optimal. Leaf size=291 \[ -\frac{245}{768 a \left (1-a^2 x^2\right )}-\frac{65}{2304 a \left (1-a^2 x^2\right )^2}-\frac{1}{216 a \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac{15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac{245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac{65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac{5 \tanh ^{-1}(a x)^4}{64 a}+\frac{245 \tanh ^{-1}(a x)^2}{768 a} \]
[Out]
-1/(216*a*(1 - a^2*x^2)^3) - 65/(2304*a*(1 - a^2*x^2)^2) - 245/(768*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(36*(1
- a^2*x^2)^3) + (65*x*ArcTanh[a*x])/(576*(1 - a^2*x^2)^2) + (245*x*ArcTanh[a*x])/(384*(1 - a^2*x^2)) + (245*A
rcTanh[a*x]^2)/(768*a) - ArcTanh[a*x]^2/(12*a*(1 - a^2*x^2)^3) - (5*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)^2) - (
15*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x]^3)/(24*(1
- a^2*x^2)^2) + (5*x*ArcTanh[a*x]^3)/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^4)/(64*a)
________________________________________________________________________________________
Rubi [A] time = 0.326792, antiderivative size = 291, normalized size of antiderivative = 1.,
number of steps used = 13, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used
= {5964, 5956, 5994, 261, 5960} \[ -\frac{245}{768 a \left (1-a^2 x^2\right )}-\frac{65}{2304 a \left (1-a^2 x^2\right )^2}-\frac{1}{216 a \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac{15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac{245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac{65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac{5 \tanh ^{-1}(a x)^4}{64 a}+\frac{245 \tanh ^{-1}(a x)^2}{768 a} \]
Antiderivative was successfully verified.
[In]
Int[ArcTanh[a*x]^3/(1 - a^2*x^2)^4,x]
[Out]
-1/(216*a*(1 - a^2*x^2)^3) - 65/(2304*a*(1 - a^2*x^2)^2) - 245/(768*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(36*(1
- a^2*x^2)^3) + (65*x*ArcTanh[a*x])/(576*(1 - a^2*x^2)^2) + (245*x*ArcTanh[a*x])/(384*(1 - a^2*x^2)) + (245*A
rcTanh[a*x]^2)/(768*a) - ArcTanh[a*x]^2/(12*a*(1 - a^2*x^2)^3) - (5*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)^2) - (
15*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x]^3)/(24*(1
- a^2*x^2)^2) + (5*x*ArcTanh[a*x]^3)/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^4)/(64*a)
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 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 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{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx &=-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac{1}{6} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^4} \, dx+\frac{5}{6} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx\\ &=-\frac{1}{216 a \left (1-a^2 x^2\right )^3}+\frac{x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac{5}{36} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac{5}{16} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac{5}{8} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{1}{216 a \left (1-a^2 x^2\right )^3}-\frac{65}{2304 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac{65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^4}{64 a}+\frac{5}{48} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\frac{15}{64} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{16} (15 a) \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{1}{216 a \left (1-a^2 x^2\right )^3}-\frac{65}{2304 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac{65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac{65 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac{65 \tanh ^{-1}(a x)^2}{768 a}-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac{15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^4}{64 a}+\frac{15}{16} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{96} (5 a) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx-\frac{1}{128} (15 a) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{1}{216 a \left (1-a^2 x^2\right )^3}-\frac{65}{2304 a \left (1-a^2 x^2\right )^2}-\frac{65}{768 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac{65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac{245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac{245 \tanh ^{-1}(a x)^2}{768 a}-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac{15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^4}{64 a}-\frac{1}{32} (15 a) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{1}{216 a \left (1-a^2 x^2\right )^3}-\frac{65}{2304 a \left (1-a^2 x^2\right )^2}-\frac{245}{768 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac{65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac{245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac{245 \tanh ^{-1}(a x)^2}{768 a}-\frac{\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac{5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac{15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac{5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac{5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac{5 \tanh ^{-1}(a x)^4}{64 a}\\ \end{align*}
Mathematica [A] time = 0.153052, size = 143, normalized size = 0.49 \[ \frac{2205 a^4 x^4-4605 a^2 x^2-144 a x \left (15 a^4 x^4-40 a^2 x^2+33\right ) \tanh ^{-1}(a x)^3-6 a x \left (735 a^4 x^4-1600 a^2 x^2+897\right ) \tanh ^{-1}(a x)+540 \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^4+9 \left (245 a^6 x^6-375 a^4 x^4-105 a^2 x^2+299\right ) \tanh ^{-1}(a x)^2+2432}{6912 a \left (a^2 x^2-1\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^4,x]
[Out]
(2432 - 4605*a^2*x^2 + 2205*a^4*x^4 - 6*a*x*(897 - 1600*a^2*x^2 + 735*a^4*x^4)*ArcTanh[a*x] + 9*(299 - 105*a^2
*x^2 - 375*a^4*x^4 + 245*a^6*x^6)*ArcTanh[a*x]^2 - 144*a*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^3 + 540
*(-1 + a^2*x^2)^3*ArcTanh[a*x]^4)/(6912*a*(-1 + a^2*x^2)^3)
________________________________________________________________________________________
Maple [C] time = 0.523, size = 3550, normalized size = 12.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arctanh(a*x)^3/(-a^2*x^2+1)^4,x)
[Out]
5/64*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^4*x^6+245/768*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*x^6+5/32*I*a^5/
(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^6-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*
arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^6-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*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*Pi*x^6-5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(I/(
(a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^6-15/32*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2
+1)+1))^3*Pi*x^4+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^4+15/64*I
*a^3/(a*x-1)^3/(a*x+1)^3*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*Pi*x^4+15/3
2*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^4+15/32*I*a/(a*x-1)^3/(a*
x+1)^3*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3*Pi*x^2-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3
*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^2-15/64*I*a/(a*x-1)^3/(a*x+1)^3*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*Pi*x^2-15/32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*
x^2+1)+1))^2*Pi*x^2+5/64*I/a/(a*x-1)^3/(a*x+1)^3*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+5/32*I/a/(a*x-1)^3/(a*x+1)^3*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))^2-5/64*I/a/(a*x-1)^3/(a*x+1)^3*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+5/64*I/a/(a*x-1)^3/(a*x+1)^3*
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))+9485/55296*a^5/(a*x-1)^3/
(a*x+1)^3*x^6+9971/55296/a/(a*x-1)^3/(a*x+1)^3-1/48/a*arctanh(a*x)^3/(a*x-1)^3-1/48/a*arctanh(a*x)^3/(a*x+1)^3
+5/32/a*arctanh(a*x)^3*ln(a*x+1)-5/16/a*arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+1/16/a*arctanh(a*x)^3/(a
*x-1)^2-5/32/a*arctanh(a*x)^3/(a*x-1)-5/32/a*arctanh(a*x)^3*ln(a*x-1)-1/16/a*arctanh(a*x)^3/(a*x+1)^2-5/32/a*a
rctanh(a*x)^3/(a*x+1)+299/768/a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2+15/64*I*a/(a*x-1)^3/(a*x+1)^3*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))*Pi*x^2-15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*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))*Pi*x^4+5/64*I*a^5/(a*x-1)^3/(a
*x+1)^3*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))*Pi*x^6-299/384/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)*x-3605/18432*a^3/(a*x-1)^3/
(a*x+1)^3*x^4-2795/18432*a/(a*x-1)^3/(a*x+1)^3*x^2-5/64/a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^4+5/64*I*a^5/(a*x-1
)^3/(a*x+1)^3*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*Pi*x^6-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(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^6+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csg
n(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*Pi*x^4+15/32*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*
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^4-15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arct
anh(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*Pi*x^4+15/
64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*csgn(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^4-15/64*I*a/(a*x-1)^3/(a*x+1)^3*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*Pi*x^2-15/32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*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+15/64*I*a/(a*x-1)^3/(a*x+1)^3*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*Pi*x^2-15/64*I*a/(a*x-1)^3/(a*x+1
)^3*arctanh(a*x)^3*csgn(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^2-5/64*I*a^5/(a*x-1
)^3/(a*x+1)^3*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*Pi*x^6-5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*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^6-5/64*I/a/(a*x-1)^3/(a*x+1)^3*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))-5/32*I/a/(a*x-1)^3/(a
*x+1)^3*Pi*arctanh(a*x)^3+5/64*I/a/(a*x-1)^3/(a*x+1)^3*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+5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2-15/3
2*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*x^4+15/32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*x^2+5/32*I*a
^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*x^6-5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^2/
(-a^2*x^2+1)+1))^3+5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3-35/256*a/(a*
x-1)^3/(a*x+1)^3*arctanh(a*x)^2*x^2-15/64*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^4*x^4-245/384*a^4/(a*x-1)^3/(a*
x+1)^3*arctanh(a*x)*x^5-125/256*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*x^4+15/64*a/(a*x-1)^3/(a*x+1)^3*arctanh
(a*x)^4*x^2+25/18*a^2/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)*x^3
________________________________________________________________________________________
Maxima [B] time = 1.11325, size = 1176, normalized size = 4.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(a*x)^3/(-a^2*x^2+1)^4,x, algorithm="maxima")
[Out]
-1/96*(2*(15*a^4*x^5 - 40*a^2*x^3 + 33*x)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1) - 15*log(a*x + 1)/a + 15*log(a
*x - 1)/a)*arctanh(a*x)^3 + 1/384*(180*a^4*x^4 - 420*a^2*x^2 - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*
x + 1)^2 + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^
2*x^2 - 1)*log(a*x - 1)^2 + 272)*a*arctanh(a*x)^2/(a^8*x^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2) + 1/27648*((8820*a^4
*x^4 - 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^4 + 540*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*lo
g(a*x + 1)^3*log(a*x - 1) - 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^4 - 18420*a^2*x^2 - 45*(49*
a^6*x^6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 - 49)*log(a*x +
1)^2 - 2205*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 + 90*(6*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)
*log(a*x - 1)^3 + 49*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1))*log(a*x + 1) + 9728)*a^2/(a^10*x^6 -
3*a^8*x^4 + 3*a^6*x^2 - a^4) - 12*(1470*a^5*x^5 - 3200*a^3*x^3 - 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(
a*x + 1)^3 + 270*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^2*log(a*x - 1) + 90*(a^6*x^6 - 3*a^4*x^4 +
3*a^2*x^2 - 1)*log(a*x - 1)^3 + 1794*a*x - 15*(49*a^6*x^6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*x
^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 - 49)*log(a*x + 1) + 735*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)
)*a*arctanh(a*x)/(a^9*x^6 - 3*a^7*x^4 + 3*a^5*x^2 - a^3))*a
________________________________________________________________________________________
Fricas [A] time = 2.06117, size = 506, normalized size = 1.74 \begin{align*} \frac{8820 \, a^{4} x^{4} + 135 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{4} - 18420 \, a^{2} x^{2} - 72 \,{\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 9 \,{\left (245 \, a^{6} x^{6} - 375 \, a^{4} x^{4} - 105 \, a^{2} x^{2} + 299\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 12 \,{\left (735 \, a^{5} x^{5} - 1600 \, a^{3} x^{3} + 897 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + 9728}{27648 \,{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(a*x)^3/(-a^2*x^2+1)^4,x, algorithm="fricas")
[Out]
1/27648*(8820*a^4*x^4 + 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1))^4 - 18420*a^2*x^2
- 72*(15*a^5*x^5 - 40*a^3*x^3 + 33*a*x)*log(-(a*x + 1)/(a*x - 1))^3 + 9*(245*a^6*x^6 - 375*a^4*x^4 - 105*a^2*x
^2 + 299)*log(-(a*x + 1)/(a*x - 1))^2 - 12*(735*a^5*x^5 - 1600*a^3*x^3 + 897*a*x)*log(-(a*x + 1)/(a*x - 1)) +
9728)/(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(atanh(a*x)**3/(-a**2*x**2+1)**4,x)
[Out]
Integral(atanh(a*x)**3/((a*x - 1)**4*(a*x + 1)**4), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(a*x)^3/(-a^2*x^2+1)^4,x, algorithm="giac")
[Out]
integrate(arctanh(a*x)^3/(a^2*x^2 - 1)^4, x)