Integrand size = 19, antiderivative size = 171 \[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=-\frac {11 x}{30}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {8 \text {arctanh}(a x)^2}{15 a}+\frac {8}{15} x \text {arctanh}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2-\frac {16 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a}-\frac {8 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{15 a} \] Output:
-11/30*x+1/30*a^2*x^3+4/15*(-a^2*x^2+1)*arctanh(a*x)/a+1/10*(-a^2*x^2+1)^2 *arctanh(a*x)/a+8/15*arctanh(a*x)^2/a+8/15*x*arctanh(a*x)^2+4/15*x*(-a^2*x ^2+1)*arctanh(a*x)^2+1/5*x*(-a^2*x^2+1)^2*arctanh(a*x)^2-16/15*arctanh(a*x )*ln(2/(-a*x+1))/a-8/15*polylog(2,1-2/(-a*x+1))/a
Time = 0.51 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.58 \[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {a x \left (-11+a^2 x^2\right )+2 (-1+a x)^3 \left (8+9 a x+3 a^2 x^2\right ) \text {arctanh}(a x)^2+\text {arctanh}(a x) \left (11-14 a^2 x^2+3 a^4 x^4-32 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )+16 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{30 a} \] Input:
Integrate[(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]
Output:
(a*x*(-11 + a^2*x^2) + 2*(-1 + a*x)^3*(8 + 9*a*x + 3*a^2*x^2)*ArcTanh[a*x] ^2 + ArcTanh[a*x]*(11 - 14*a^2*x^2 + 3*a^4*x^4 - 32*Log[1 + E^(-2*ArcTanh[ a*x])]) + 16*PolyLog[2, -E^(-2*ArcTanh[a*x])])/(30*a)
Time = 0.88 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6506, 2009, 6506, 24, 6436, 6546, 6470, 2849, 2752}
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.
| \(\displaystyle \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6506 |
| \(\displaystyle \frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2dx-\frac {1}{10} \int \left (1-a^2 x^2\right )dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
| \(\Big \downarrow \) 6506 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int \text {arctanh}(a x)^2dx-\frac {\int 1dx}{3}+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int \text {arctanh}(a x)^2dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\int \frac {\text {arctanh}(a x)}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\) |
Input:
Int[(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]
Output:
(-x + (a^2*x^3)/3)/10 + ((1 - a^2*x^2)^2*ArcTanh[a*x])/(10*a) + (x*(1 - a^ 2*x^2)^2*ArcTanh[a*x]^2)/5 + (4*(-1/3*x + ((1 - a^2*x^2)*ArcTanh[a*x])/(3* a) + (x*(1 - a^2*x^2)*ArcTanh[a*x]^2)/3 + (2*(x*ArcTanh[a*x]^2 - 2*a*(-1/2 *ArcTanh[a*x]^2/a^2 + ((ArcTanh[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)))/3))/5
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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])
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]
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]
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]
Time = 0.46 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{2} a x +\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{10}-\frac {7 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{15}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {a^{3} x^{3}}{30}-\frac {11 a x}{30}-\frac {11 \ln \left (a x -1\right )}{60}+\frac {11 \ln \left (a x +1\right )}{60}+\frac {2 \ln \left (a x -1\right )^{2}}{15}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {2 \ln \left (a x +1\right )^{2}}{15}}{a}\) | \(188\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}+\operatorname {arctanh}\left (a x \right )^{2} a x +\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{10}-\frac {7 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{15}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {a^{3} x^{3}}{30}-\frac {11 a x}{30}-\frac {11 \ln \left (a x -1\right )}{60}+\frac {11 \ln \left (a x +1\right )}{60}+\frac {2 \ln \left (a x -1\right )^{2}}{15}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {2 \ln \left (a x +1\right )^{2}}{15}}{a}\) | \(188\) |
| parts | \(\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{4} x^{5}}{5}-\frac {2 x^{3} a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{3}+x \operatorname {arctanh}\left (a x \right )^{2}+\frac {a^{3} \operatorname {arctanh}\left (a x \right ) x^{4}}{10}-\frac {7 x^{2} \operatorname {arctanh}\left (a x \right ) a}{15}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15 a}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15 a}-\frac {-4 \ln \left (a x -1\right )^{2}+16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )-8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )+4 \ln \left (a x +1\right )^{2}-a^{3} x^{3}+11 a x +\frac {11 \ln \left (a x -1\right )}{2}-\frac {11 \ln \left (a x +1\right )}{2}}{30 a}\) | \(193\) |
| risch | \(-\frac {11 x}{30}+\frac {a^{3} \ln \left (a x +1\right ) x^{4}}{20}-\frac {a^{3} \ln \left (-a x +1\right ) x^{4}}{20}+\frac {a^{2} x^{3}}{30}+\frac {7 a \ln \left (-a x +1\right ) x^{2}}{30}+\frac {7 \ln \left (-a x +1\right ) \ln \left (a x +1\right )}{30 a}-\frac {7 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{15 a}+\frac {7 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15 a}-\frac {7 a \ln \left (a x +1\right ) x^{2}}{30}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2 a}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {x \ln \left (-a x +1\right )}{2}+\frac {a^{4} \ln \left (-a x +1\right )^{2} x^{5}}{20}-\frac {19 \ln \left (a x +1\right )}{60 a}-\frac {94 \ln \left (a x -1\right )}{225 a}-\frac {239 \ln \left (-a x +1\right )}{900 a}-\frac {\ln \left (a x +1\right ) x}{2}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15 a}+\frac {a^{4} \ln \left (a x +1\right )^{2} x^{5}}{20}+\frac {\ln \left (-a x +1\right )^{2} x}{4}-\frac {2 \ln \left (-a x +1\right )^{2}}{15 a}+\frac {\ln \left (a x +1\right )^{2} x}{4}+\frac {2 \ln \left (a x +1\right )^{2}}{15 a}-\frac {a^{4} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{10}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{3}}{6}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{3}}{6}-\frac {3739}{6750 a}-\frac {\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right ) \ln \left (-a x +1\right )}{2 a}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{3}}{3}\) | \(418\) |
Input:
int((-a^2*x^2+1)^2*arctanh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a*(1/5*arctanh(a*x)^2*a^5*x^5-2/3*arctanh(a*x)^2*a^3*x^3+arctanh(a*x)^2* a*x+1/10*a^4*x^4*arctanh(a*x)-7/15*a^2*x^2*arctanh(a*x)+8/15*arctanh(a*x)* ln(a*x-1)+8/15*arctanh(a*x)*ln(a*x+1)+1/30*a^3*x^3-11/30*a*x-11/60*ln(a*x- 1)+11/60*ln(a*x+1)+2/15*ln(a*x-1)^2-8/15*dilog(1/2*a*x+1/2)-4/15*ln(a*x-1) *ln(1/2*a*x+1/2)+4/15*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-2/15*ln (a*x+1)^2)
\[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int { {\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="fricas")
Output:
integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2, x)
\[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \] Input:
integrate((-a**2*x**2+1)**2*atanh(a*x)**2,x)
Output:
Integral((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2, x)
Time = 0.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02 \[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} - 22 \, a x - 8 \, \log \left (a x + 1\right )^{2} + 16 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 8 \, \log \left (a x - 1\right )^{2} - 11 \, \log \left (a x - 1\right )}{a^{3}} - \frac {32 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}} + \frac {11 \, \log \left (a x + 1\right )}{a^{3}}\right )} + \frac {1}{30} \, {\left (3 \, a^{2} x^{4} - 14 \, x^{2} + \frac {16 \, \log \left (a x + 1\right )}{a^{2}} + \frac {16 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {1}{15} \, {\left (3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, x\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="maxima")
Output:
1/60*a^2*((2*a^3*x^3 - 22*a*x - 8*log(a*x + 1)^2 + 16*log(a*x + 1)*log(a*x - 1) + 8*log(a*x - 1)^2 - 11*log(a*x - 1))/a^3 - 32*(log(a*x - 1)*log(1/2 *a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^3 + 11*log(a*x + 1)/a^3) + 1/30*(3* a^2*x^4 - 14*x^2 + 16*log(a*x + 1)/a^2 + 16*log(a*x - 1)/a^2)*a*arctanh(a* x) + 1/15*(3*a^4*x^5 - 10*a^2*x^3 + 15*x)*arctanh(a*x)^2
\[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int { {\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="giac")
Output:
integrate((a^2*x^2 - 1)^2*arctanh(a*x)^2, x)
Timed out. \[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int {\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2 \,d x \] Input:
int(atanh(a*x)^2*(a^2*x^2 - 1)^2,x)
Output:
int(atanh(a*x)^2*(a^2*x^2 - 1)^2, x)
\[ \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {6 \mathit {atanh} \left (a x \right )^{2} a^{5} x^{5}-20 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}+30 \mathit {atanh} \left (a x \right )^{2} a x +3 \mathit {atanh} \left (a x \right ) a^{4} x^{4}-14 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+11 \mathit {atanh} \left (a x \right )+32 \left (\int \frac {\mathit {atanh} \left (a x \right ) x}{a^{2} x^{2}-1}d x \right ) a^{2}+a^{3} x^{3}-11 a x}{30 a} \] Input:
int((-a^2*x^2+1)^2*atanh(a*x)^2,x)
Output:
(6*atanh(a*x)**2*a**5*x**5 - 20*atanh(a*x)**2*a**3*x**3 + 30*atanh(a*x)**2 *a*x + 3*atanh(a*x)*a**4*x**4 - 14*atanh(a*x)*a**2*x**2 + 11*atanh(a*x) + 32*int((atanh(a*x)*x)/(a**2*x**2 - 1),x)*a**2 + a**3*x**3 - 11*a*x)/(30*a)