\(\int (1-a^2 x^2) \text {arctanh}(a x)^3 \, dx\) [183]

Optimal result

Integrand size = 17, antiderivative size = 157 \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=-x \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}+\frac {2 \text {arctanh}(a x)^3}{3 a}+\frac {2}{3} x \text {arctanh}(a x)^3+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3-\frac {2 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85 \[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=-\frac {6 a x \text {arctanh}(a x)-3 \text {arctanh}(a x)^2+3 a^2 x^2 \text {arctanh}(a x)^2+4 \text {arctanh}(a x)^3-6 a x \text {arctanh}(a x)^3+2 a^3 x^3 \text {arctanh}(a x)^3+12 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+3 \log \left (1-a^2 x^2\right )-12 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{6 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6506, 6436, 240, 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.

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

\(\Big \downarrow \) 6506

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

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 6506

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x 
_Symbol] :> Simp[b*p*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2* 
q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^p/(2*q + 1)), x] 
 + Simp[2*d*(q/(2*q + 1))   Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, 
 x], x] - Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1)))   Int[(d + e*x^2)^(q - 1)* 
(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c 
^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 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 = 2.94 (sec) , antiderivative size = 772, normalized size of antiderivative = 4.92

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{3} a x -\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{3}-\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 )^{2}}{2}+\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )-\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (2\right )+\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 )^{2}}{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 )^{2}}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2}+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 )^{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}}{\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 )^{2}}{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 )^{2}}{2}-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 )^{2}-i \pi \operatorname {arctanh}\left (a x \right )^{2}-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 )^{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 )^{2}}{2}}{a}\)	\(772\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{3} a x -\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{3}-\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 )^{2}}{2}+\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )-\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (2\right )+\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 )^{2}}{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 )^{2}}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2}+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 )^{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}}{\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 )^{2}}{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 )^{2}}{2}-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 )^{2}-i \pi \operatorname {arctanh}\left (a x \right )^{2}-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 )^{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 )^{2}}{2}}{a}\)	\(772\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{3}}{3}+x \operatorname {arctanh}\left (a x \right )^{3}-\frac {a \operatorname {arctanh}\left (a x \right )^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{a}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{a}+\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 )+\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{3}-\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 )^{2}}{2}+\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )-\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (2\right )+\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 )^{2}}{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 )^{2}}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2}+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 )^{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}}{\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 )^{2}}{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 )^{2}}{2}-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 )^{2}-i \pi \operatorname {arctanh}\left (a x \right )^{2}-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 )^{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 )^{2}}{2}}{a}\)	\(776\)

Input:

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

Output:

1/a*(-1/3*arctanh(a*x)^3*a^3*x^3+arctanh(a*x)^3*a*x-1/2*a^2*x^2*arctanh(a* 
x)^2+arctanh(a*x)^2*ln(a*x-1)+arctanh(a*x)^2*ln(a*x+1)-2*arctanh(a*x)*poly 
log(2,-(a*x+1)^2/(-a^2*x^2+1))+polylog(3,-(a*x+1)^2/(-a^2*x^2+1))-2*arctan 
h(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+2/3*arctanh(a*x)^3-1/2*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)^ 
2+ln((a*x+1)^2/(-a^2*x^2+1)+1)-(a*x+1)*arctanh(a*x)-2*arctanh(a*x)^2*ln(2) 
+1/2*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)^2-1/2*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2 
-1))^3*arctanh(a*x)^2+1/2*arctanh(a*x)^2+I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2- 
1)+1))^2*arctanh(a*x)^2-1/2*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)^2-1/2*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) 
^2-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)^2-I*Pi*arctanh(a*x)^2-I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1) 
)^3*arctanh(a*x)^2+1/2*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)^2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/48*(2*a^3*x^3 - 3*a^2*x^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2)*log(a*x + 1 
))*log(-a*x + 1)^2/a - 1/8*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a 
*x + 1) - 6)*(a*x - 1)/a + 1/864*(4*(9*log(-a*x + 1)^3 - 9*log(-a*x + 1)^2 
 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 27*(4*log(-a*x + 1)^3 - 6*log(-a*x + 
 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 108*(log(-a*x + 1)^3 - 3*log(-a 
*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a + 1/8*integrate(-1/3*(3*(a^3 
*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)^3 + (2*a^3*x^3 - 3*a^2*x^2 - 9*(a^3 
*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)^2 - 12*a*x - 6*(a^3*x^3 - 3*a*x - 2 
)*log(a*x + 1))*log(-a*x + 1))/(a*x - 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3 \, dx=\frac {-2 \mathit {atanh} \left (a x \right )^{3} a^{3} x^{3}+6 \mathit {atanh} \left (a x \right )^{3} a x -3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+3 \mathit {atanh} \left (a x \right )^{2}-6 \mathit {atanh} \left (a x \right ) a x -6 \mathit {atanh} \left (a x \right )+12 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2} x}{a^{2} x^{2}-1}d x \right ) a^{2}-6 \,\mathrm {log}\left (a^{2} x -a \right )}{6 a} \] Input:

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

Output:

( - 2*atanh(a*x)**3*a**3*x**3 + 6*atanh(a*x)**3*a*x - 3*atanh(a*x)**2*a**2 
*x**2 + 3*atanh(a*x)**2 - 6*atanh(a*x)*a*x - 6*atanh(a*x) + 12*int((atanh( 
a*x)**2*x)/(a**2*x**2 - 1),x)*a**2 - 6*log(a**2*x - a))/(6*a)