Integrand size = 22, antiderivative size = 178 \[ \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=-\frac {x}{210 a^2}-\frac {17 x^3}{630}+\frac {a^2 x^5}{105}+\frac {\text {arctanh}(a x)}{210 a^3}+\frac {8 x^2 \text {arctanh}(a x)}{105 a}-\frac {9}{70} a x^4 \text {arctanh}(a x)+\frac {1}{21} a^3 x^6 \text {arctanh}(a x)+\frac {8 \text {arctanh}(a x)^2}{105 a^3}+\frac {1}{3} x^3 \text {arctanh}(a x)^2-\frac {2}{5} a^2 x^5 \text {arctanh}(a x)^2+\frac {1}{7} a^4 x^7 \text {arctanh}(a x)^2-\frac {16 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{105 a^3}-\frac {8 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{105 a^3} \]
-1/210*x/a^2-17/630*x^3+1/105*a^2*x^5+1/210*arctanh(a*x)/a^3+8/105*x^2*arc tanh(a*x)/a-9/70*a*x^4*arctanh(a*x)+1/21*a^3*x^6*arctanh(a*x)+8/105*arctan h(a*x)^2/a^3+1/3*x^3*arctanh(a*x)^2-2/5*a^2*x^5*arctanh(a*x)^2+1/7*a^4*x^7 *arctanh(a*x)^2-16/105*arctanh(a*x)*ln(2/(-a*x+1))/a^3-8/105*polylog(2,1-2 /(-a*x+1))/a^3
Time = 0.86 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68 \[ \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {a x \left (-3-17 a^2 x^2+6 a^4 x^4\right )+6 \left (-8+35 a^3 x^3-42 a^5 x^5+15 a^7 x^7\right ) \text {arctanh}(a x)^2+\text {arctanh}(a x) \left (3+48 a^2 x^2-81 a^4 x^4+30 a^6 x^6-96 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )+48 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{630 a^3} \]
(a*x*(-3 - 17*a^2*x^2 + 6*a^4*x^4) + 6*(-8 + 35*a^3*x^3 - 42*a^5*x^5 + 15* a^7*x^7)*ArcTanh[a*x]^2 + ArcTanh[a*x]*(3 + 48*a^2*x^2 - 81*a^4*x^4 + 30*a ^6*x^6 - 96*Log[1 + E^(-2*ArcTanh[a*x])]) + 48*PolyLog[2, -E^(-2*ArcTanh[a *x])])/(630*a^3)
Time = 0.97 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6574, 2009}
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.
| \(\displaystyle \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \int \left (a^4 x^6 \text {arctanh}(a x)^2-2 a^2 x^4 \text {arctanh}(a x)^2+x^2 \text {arctanh}(a x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} a^4 x^7 \text {arctanh}(a x)^2+\frac {1}{21} a^3 x^6 \text {arctanh}(a x)+\frac {8 \text {arctanh}(a x)^2}{105 a^3}+\frac {\text {arctanh}(a x)}{210 a^3}-\frac {16 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{105 a^3}-\frac {8 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{105 a^3}-\frac {2}{5} a^2 x^5 \text {arctanh}(a x)^2+\frac {a^2 x^5}{105}-\frac {x}{210 a^2}-\frac {9}{70} a x^4 \text {arctanh}(a x)+\frac {1}{3} x^3 \text {arctanh}(a x)^2+\frac {8 x^2 \text {arctanh}(a x)}{105 a}-\frac {17 x^3}{630}\) |
-1/210*x/a^2 - (17*x^3)/630 + (a^2*x^5)/105 + ArcTanh[a*x]/(210*a^3) + (8* x^2*ArcTanh[a*x])/(105*a) - (9*a*x^4*ArcTanh[a*x])/70 + (a^3*x^6*ArcTanh[a *x])/21 + (8*ArcTanh[a*x]^2)/(105*a^3) + (x^3*ArcTanh[a*x]^2)/3 - (2*a^2*x ^5*ArcTanh[a*x]^2)/5 + (a^4*x^7*ArcTanh[a*x]^2)/7 - (16*ArcTanh[a*x]*Log[2 /(1 - a*x)])/(105*a^3) - (8*PolyLog[2, 1 - 2/(1 - a*x)])/(105*a^3)
3.3.5.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
Time = 0.32 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{21}-\frac {9 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{70}+\frac {8 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{105}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{105}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{105}+\frac {a^{5} x^{5}}{105}-\frac {17 a^{3} x^{3}}{630}-\frac {a x}{210}-\frac {\ln \left (a x -1\right )}{420}+\frac {\ln \left (a x +1\right )}{420}+\frac {2 \ln \left (a x -1\right )^{2}}{105}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{105}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{105}-\frac {2 \ln \left (a x +1\right )^{2}}{105}}{a^{3}}\) | \(213\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{21}-\frac {9 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{70}+\frac {8 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{105}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{105}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{105}+\frac {a^{5} x^{5}}{105}-\frac {17 a^{3} x^{3}}{630}-\frac {a x}{210}-\frac {\ln \left (a x -1\right )}{420}+\frac {\ln \left (a x +1\right )}{420}+\frac {2 \ln \left (a x -1\right )^{2}}{105}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{105}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{105}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{105}-\frac {2 \ln \left (a x +1\right )^{2}}{105}}{a^{3}}\) | \(213\) |
| parts | \(\frac {a^{4} x^{7} \operatorname {arctanh}\left (a x \right )^{2}}{7}-\frac {2 a^{2} x^{5} \operatorname {arctanh}\left (a x \right )^{2}}{5}+\frac {x^{3} \operatorname {arctanh}\left (a x \right )^{2}}{3}+\frac {a^{3} x^{6} \operatorname {arctanh}\left (a x \right )}{21}-\frac {9 a \,x^{4} \operatorname {arctanh}\left (a x \right )}{70}+\frac {8 x^{2} \operatorname {arctanh}\left (a x \right )}{105 a}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{105 a^{3}}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{105 a^{3}}-\frac {-4 \ln \left (a x -1\right )^{2}+16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )-8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )+4 \ln \left (a x +1\right )^{2}-2 a^{5} x^{5}+\frac {17 a^{3} x^{3}}{3}+a x +\frac {\ln \left (a x -1\right )}{2}-\frac {\ln \left (a x +1\right )}{2}}{210 a^{3}}\) | \(215\) |
| risch | \(-\frac {177151}{2315250 a^{3}}+\frac {a^{2} x^{5}}{105}-\frac {x}{210 a^{2}}-\frac {x^{3} \ln \left (-a x +1\right )}{18}+\frac {2 \ln \left (a x +1\right )^{2}}{105 a^{3}}-\frac {\ln \left (a x +1\right ) x^{3}}{18}+\frac {\ln \left (a x +1\right )^{2} x^{3}}{12}-\frac {43 \ln \left (a x +1\right )}{315 a^{3}}+\frac {\ln \left (-a x +1\right )^{2} x^{3}}{12}-\frac {2617 \ln \left (-a x +1\right )}{11025 a^{3}}-\frac {2 \ln \left (-a x +1\right )^{2}}{105 a^{3}}-\frac {17 x^{3}}{630}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{5}-\frac {a^{4} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{7}}{14}+\frac {\left (a x +1\right )^{3} \ln \left (a x +1\right )}{18 a^{3}}-\frac {\left (\left (-\frac {1}{9}+\frac {\ln \left (a x +1\right )}{3}\right ) \left (a x +1\right )^{3}+\left (\frac {1}{2}-\ln \left (a x +1\right )\right ) \left (a x +1\right )^{2}+\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right )\right ) \ln \left (-a x +1\right )}{2 a^{3}}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{6 a^{3}}+\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 )}{3 a^{3}}+\frac {9 \ln \left (-a x +1\right ) \ln \left (a x +1\right )}{70 a^{3}}-\frac {9 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{35 a^{3}}+\frac {9 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a^{3}}-\frac {\left (a x +1\right )^{2} \ln \left (a x +1\right )}{12 a^{3}}-\frac {778 \ln \left (a x -1\right )}{11025 a^{3}}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{105 a^{3}}-\frac {\ln \left (a x +1\right ) x}{6 a^{2}}+\frac {a^{4} \ln \left (a x +1\right )^{2} x^{7}}{28}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{5}}{10}+\frac {a^{3} \ln \left (a x +1\right ) x^{6}}{42}-\frac {9 a \ln \left (a x +1\right ) x^{4}}{140}-\frac {19 \ln \left (a x +1\right ) x^{2}}{420 a}-\frac {\ln \left (-a x +1\right ) x}{6 a^{2}}+\frac {a^{4} \ln \left (-a x +1\right )^{2} x^{7}}{28}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{5}}{10}-\frac {a^{3} \ln \left (-a x +1\right ) x^{6}}{42}+\frac {9 a \ln \left (-a x +1\right ) x^{4}}{140}+\frac {19 \ln \left (-a x +1\right ) x^{2}}{420 a}\) | \(563\) |
1/a^3*(1/7*arctanh(a*x)^2*a^7*x^7-2/5*arctanh(a*x)^2*a^5*x^5+1/3*arctanh(a *x)^2*a^3*x^3+1/21*arctanh(a*x)*a^6*x^6-9/70*a^4*x^4*arctanh(a*x)+8/105*a^ 2*x^2*arctanh(a*x)+8/105*arctanh(a*x)*ln(a*x-1)+8/105*arctanh(a*x)*ln(a*x+ 1)+1/105*a^5*x^5-17/630*a^3*x^3-1/210*a*x-1/420*ln(a*x-1)+1/420*ln(a*x+1)+ 2/105*ln(a*x-1)^2-8/105*dilog(1/2*a*x+1/2)-4/105*ln(a*x-1)*ln(1/2*a*x+1/2) +4/105*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-2/105*ln(a*x+1)^2)
\[ \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int { {\left (a^{2} x^{2} - 1\right )}^{2} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \]
\[ \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int x^{2} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Time = 0.18 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11 \[ \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {1}{1260} \, a^{2} {\left (\frac {12 \, a^{5} x^{5} - 34 \, a^{3} x^{3} - 6 \, a x - 24 \, \log \left (a x + 1\right )^{2} + 48 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 24 \, \log \left (a x - 1\right )^{2} - 3 \, \log \left (a x - 1\right )}{a^{5}} - \frac {96 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} + \frac {3 \, \log \left (a x + 1\right )}{a^{5}}\right )} + \frac {1}{210} \, a {\left (\frac {10 \, a^{4} x^{6} - 27 \, a^{2} x^{4} + 16 \, x^{2}}{a^{2}} + \frac {16 \, \log \left (a x + 1\right )}{a^{4}} + \frac {16 \, \log \left (a x - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{105} \, {\left (15 \, a^{4} x^{7} - 42 \, a^{2} x^{5} + 35 \, x^{3}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
1/1260*a^2*((12*a^5*x^5 - 34*a^3*x^3 - 6*a*x - 24*log(a*x + 1)^2 + 48*log( a*x + 1)*log(a*x - 1) + 24*log(a*x - 1)^2 - 3*log(a*x - 1))/a^5 - 96*(log( a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^5 + 3*log(a*x + 1)/ a^5) + 1/210*a*((10*a^4*x^6 - 27*a^2*x^4 + 16*x^2)/a^2 + 16*log(a*x + 1)/a ^4 + 16*log(a*x - 1)/a^4)*arctanh(a*x) + 1/105*(15*a^4*x^7 - 42*a^2*x^5 + 35*x^3)*arctanh(a*x)^2
\[ \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int { {\left (a^{2} x^{2} - 1\right )}^{2} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \]
Timed out. \[ \int x^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2 \,d x \]