\(\int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx\) [242]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=-\frac {\text {arctanh}(a x)^3}{a^3}-\frac {x \text {arctanh}(a x)^3}{a^2}+\frac {\text {arctanh}(a x)^4}{4 a^3}+\frac {3 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3} \] Output:

-arctanh(a*x)^3/a^3-x*arctanh(a*x)^3/a^2+1/4*arctanh(a*x)^4/a^3+3*arctanh( 
a*x)^2*ln(2/(-a*x+1))/a^3+3*arctanh(a*x)*polylog(2,1-2/(-a*x+1))/a^3-3/2*p 
olylog(3,1-2/(-a*x+1))/a^3
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\frac {\text {arctanh}(a x)^2 \left ((4-4 a x) \text {arctanh}(a x)+\text {arctanh}(a x)^2+12 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-12 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{4 a^3} \] Input:

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

Output:

(ArcTanh[a*x]^2*((4 - 4*a*x)*ArcTanh[a*x] + ArcTanh[a*x]^2 + 12*Log[1 + E^ 
(-2*ArcTanh[a*x])]) - 12*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] - 6 
*PolyLog[3, -E^(-2*ArcTanh[a*x])])/(4*a^3)
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6542, 6436, 6510, 6546, 6470, 6620, 7164}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx\)

\(\Big \downarrow \) 6542

\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx}{a^2}-\frac {\int \text {arctanh}(a x)^3dx}{a^2}\)

\(\Big \downarrow \) 6436

\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx}{a^2}-\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}\)

\(\Big \downarrow \) 6510

\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\)

\(\Big \downarrow \) 6620

\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {\text {arctanh}(a x)^4}{4 a^3}-\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a^2}\)

Input:

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

Output:

ArcTanh[a*x]^4/(4*a^3) - (x*ArcTanh[a*x]^3 - 3*a*(-1/3*ArcTanh[a*x]^3/a^2 
+ ((ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a - 2*(-1/2*(ArcTanh[a*x]*PolyLog[2, 
1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/(4*a)))/a))/a^2
 

Defintions of rubi rules used

rule 6436

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTanh[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 1])
 

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6510

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b 
, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
 

rule 6542

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e   Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* 
x])^p, x], x] - Simp[d*(f^2/e)   Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ 
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 
 1]
 

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 6620

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*(p/2)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( 
d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d 
 + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 16.36 (sec) , antiderivative size = 736, normalized size of antiderivative = 7.15

method	result	size
derivativedivides	\(\frac {-\operatorname {arctanh}\left (a x \right )^{3} a x -\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\operatorname {arctanh}\left (a x \right )^{3}-\frac {3 \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3}}\)	\(736\)
default	\(\frac {-\operatorname {arctanh}\left (a x \right )^{3} a x -\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2}-\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{4}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}-\operatorname {arctanh}\left (a x \right )^{3}-\frac {3 \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{4}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3}}\)	\(736\)
parts	\(-\frac {x \operatorname {arctanh}\left (a x \right )^{3}}{a^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{2 a^{3}}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x -1\right )}{2 a^{3}}-\frac {3 a \left (\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{4}}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{6 a^{4}}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{4}}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{4}}+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{4}}-\frac {i \pi \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{4}}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}+\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{4}}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{a^{4}}-\frac {2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{4}}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{4}}+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{4}}\right )}{2}\)	\(791\)

Input:

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

Output:

1/a^3*(-arctanh(a*x)^3*a*x-1/2*arctanh(a*x)^3*ln(a*x-1)+1/2*arctanh(a*x)^3 
*ln(a*x+1)-arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*arctanh(a*x)^ 
4+1/4*I*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*arctanh(a*x)^3-1/4*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)^3-1/4*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)^3-1/2*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*cs 
gn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^3+1/4*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/ 
(-(a*x+1)^2/(a^2*x^2-1)+1))*arctanh(a*x)^3+1/2*I*Pi*arctanh(a*x)^3-1/2*I*P 
i*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)^3+1/2*I*Pi*csgn(I/(-(a 
*x+1)^2/(a^2*x^2-1)+1))^3*arctanh(a*x)^3-1/4*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-arctanh(a*x)^3-3/2*polyl 
og(3,-(a*x+1)^2/(-a^2*x^2+1))-1/4*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arc 
tanh(a*x)^3+3*arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)+1)+3*arctanh(a*x)*p 
olylog(2,-(a*x+1)^2/(-a^2*x^2+1)))
 

Fricas [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=- \int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1} \,d x } \] Input:

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

Output:

1/64*(4*(2*a*x - log(a*x + 1) - 2)*log(-a*x + 1)^3 + log(-a*x + 1)^4 - 6*( 
4*(a*x + 1)*log(a*x + 1) - log(a*x + 1)^2)*log(-a*x + 1)^2)/a^3 + 1/8*inte 
grate(-1/2*(2*a^2*x^2*log(a*x + 1)^3 - 3*((2*a^2*x^2 - a*x - 1)*log(a*x + 
1)^2 + 4*(a^2*x^2 + 2*a*x + 1)*log(a*x + 1))*log(-a*x + 1))/(a^4*x^2 - a^2 
), x)
 

Giac [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=-\int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{a^2\,x^2-1} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{1-a^2 x^2} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{3} x^{2}}{a^{2} x^{2}-1}d x \right ) \] Input:

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

Output:

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