Integrand size = 22, antiderivative size = 157 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx=-\frac {a^2}{30 x^3}+\frac {11 a^4}{30 x}-\frac {11}{30} a^5 \text {arctanh}(a x)-\frac {a \text {arctanh}(a x)}{10 x^4}+\frac {7 a^3 \text {arctanh}(a x)}{15 x^2}+\frac {8}{15} a^5 \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{5 x^5}+\frac {2 a^2 \text {arctanh}(a x)^2}{3 x^3}-\frac {a^4 \text {arctanh}(a x)^2}{x}+\frac {16}{15} a^5 \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {8}{15} a^5 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
-1/30*a^2/x^3+11/30*a^4/x-11/30*a^5*arctanh(a*x)-1/10*a*arctanh(a*x)/x^4+7 /15*a^3*arctanh(a*x)/x^2+8/15*a^5*arctanh(a*x)^2-1/5*arctanh(a*x)^2/x^5+2/ 3*a^2*arctanh(a*x)^2/x^3-a^4*arctanh(a*x)^2/x+16/15*a^5*arctanh(a*x)*ln(2- 2/(a*x+1))-8/15*a^5*polylog(2,-1+2/(a*x+1))
Time = 0.57 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx=\frac {a^2 x^2 \left (-1+11 a^2 x^2\right )+2 (-1+a x)^3 \left (3+9 a x+8 a^2 x^2\right ) \text {arctanh}(a x)^2+a x \text {arctanh}(a x) \left (-3+14 a^2 x^2-11 a^4 x^4+32 a^4 x^4 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )-16 a^5 x^5 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )}{30 x^5} \]
(a^2*x^2*(-1 + 11*a^2*x^2) + 2*(-1 + a*x)^3*(3 + 9*a*x + 8*a^2*x^2)*ArcTan h[a*x]^2 + a*x*ArcTanh[a*x]*(-3 + 14*a^2*x^2 - 11*a^4*x^4 + 32*a^4*x^4*Log [1 - E^(-2*ArcTanh[a*x])]) - 16*a^5*x^5*PolyLog[2, E^(-2*ArcTanh[a*x])])/( 30*x^5)
Time = 0.82 (sec) , antiderivative size = 157, 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.
| \(\displaystyle \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \int \left (\frac {a^4 \text {arctanh}(a x)^2}{x^2}-\frac {2 a^2 \text {arctanh}(a x)^2}{x^4}+\frac {\text {arctanh}(a x)^2}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8}{15} a^5 \text {arctanh}(a x)^2-\frac {11}{30} a^5 \text {arctanh}(a x)+\frac {16}{15} a^5 \text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {8}{15} a^5 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {a^4 \text {arctanh}(a x)^2}{x}+\frac {11 a^4}{30 x}+\frac {7 a^3 \text {arctanh}(a x)}{15 x^2}+\frac {2 a^2 \text {arctanh}(a x)^2}{3 x^3}-\frac {a^2}{30 x^3}-\frac {\text {arctanh}(a x)^2}{5 x^5}-\frac {a \text {arctanh}(a x)}{10 x^4}\) |
-1/30*a^2/x^3 + (11*a^4)/(30*x) - (11*a^5*ArcTanh[a*x])/30 - (a*ArcTanh[a* x])/(10*x^4) + (7*a^3*ArcTanh[a*x])/(15*x^2) + (8*a^5*ArcTanh[a*x]^2)/15 - ArcTanh[a*x]^2/(5*x^5) + (2*a^2*ArcTanh[a*x]^2)/(3*x^3) - (a^4*ArcTanh[a* x]^2)/x + (16*a^5*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)])/15 - (8*a^5*PolyLog[2 , -1 + 2/(1 + a*x)])/15
3.3.13.3.1 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 = 0.51 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(a^{5} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{5 a^{5} x^{5}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{a x}+\frac {2 \operatorname {arctanh}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{10 a^{4} x^{4}}+\frac {7 \,\operatorname {arctanh}\left (a x \right )}{15 a^{2} x^{2}}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{15}-\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}-\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {1}{30 a^{3} x^{3}}+\frac {11}{30 a x}+\frac {11 \ln \left (a x -1\right )}{60}-\frac {11 \ln \left (a x +1\right )}{60}-\frac {8 \operatorname {dilog}\left (a x \right )}{15}-\frac {8 \operatorname {dilog}\left (a x +1\right )}{15}-\frac {8 \ln \left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {2 \ln \left (a x -1\right )^{2}}{15}+\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {2 \ln \left (a x +1\right )^{2}}{15}\right )\) | \(233\) |
| default | \(a^{5} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{5 a^{5} x^{5}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{a x}+\frac {2 \operatorname {arctanh}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{10 a^{4} x^{4}}+\frac {7 \,\operatorname {arctanh}\left (a x \right )}{15 a^{2} x^{2}}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{15}-\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}-\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {1}{30 a^{3} x^{3}}+\frac {11}{30 a x}+\frac {11 \ln \left (a x -1\right )}{60}-\frac {11 \ln \left (a x +1\right )}{60}-\frac {8 \operatorname {dilog}\left (a x \right )}{15}-\frac {8 \operatorname {dilog}\left (a x +1\right )}{15}-\frac {8 \ln \left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {2 \ln \left (a x -1\right )^{2}}{15}+\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {2 \ln \left (a x +1\right )^{2}}{15}\right )\) | \(233\) |
| parts | \(-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{5 x^{5}}+\frac {2 a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{3 x^{3}}-\frac {a^{4} \operatorname {arctanh}\left (a x \right )^{2}}{x}-\frac {a \,\operatorname {arctanh}\left (a x \right )}{10 x^{4}}+\frac {7 a^{3} \operatorname {arctanh}\left (a x \right )}{15 x^{2}}+\frac {16 a^{5} \operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )}{15}-\frac {8 a^{5} \operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}-\frac {8 a^{5} \operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {a^{5} \left (-\frac {1}{a^{3} x^{3}}+\frac {11}{a x}+\frac {11 \ln \left (a x -1\right )}{2}-\frac {11 \ln \left (a x +1\right )}{2}-16 \operatorname {dilog}\left (a x \right )-16 \operatorname {dilog}\left (a x +1\right )-16 \ln \left (a x \right ) \ln \left (a x +1\right )-4 \ln \left (a x -1\right )^{2}+16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )-8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )+4 \ln \left (a x +1\right )^{2}\right )}{30}\) | \(239\) |
a^5*(-1/5*arctanh(a*x)^2/a^5/x^5-arctanh(a*x)^2/a/x+2/3*arctanh(a*x)^2/a^3 /x^3-1/10*arctanh(a*x)/a^4/x^4+7/15*arctanh(a*x)/a^2/x^2+16/15*arctanh(a*x )*ln(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)-8/15*dilog(a*x)-8/15*dilog( a*x+1)-8/15*ln(a*x)*ln(a*x+1)-2/15*ln(a*x-1)^2+8/15*dilog(1/2*a*x+1/2)+4/1 5*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)
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{6}} \,d x } \]
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx=\int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{6}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.52 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx=\frac {1}{60} \, {\left (32 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a^{3} - 32 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a^{3} + 32 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a^{3} - 11 \, a^{3} \log \left (a x + 1\right ) + 11 \, a^{3} \log \left (a x - 1\right ) + \frac {2 \, {\left (4 \, a^{3} x^{3} \log \left (a x + 1\right )^{2} - 8 \, a^{3} x^{3} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 4 \, a^{3} x^{3} \log \left (a x - 1\right )^{2} + 11 \, a^{2} x^{2} - 1\right )}}{x^{3}}\right )} a^{2} - \frac {1}{30} \, {\left (16 \, a^{4} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{4} \log \left (x^{2}\right ) - \frac {14 \, a^{2} x^{2} - 3}{x^{4}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {{\left (15 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 3\right )} \operatorname {artanh}\left (a x\right )^{2}}{15 \, x^{5}} \]
1/60*(32*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a^3 - 3 2*(log(a*x + 1)*log(x) + dilog(-a*x))*a^3 + 32*(log(-a*x + 1)*log(x) + dil og(a*x))*a^3 - 11*a^3*log(a*x + 1) + 11*a^3*log(a*x - 1) + 2*(4*a^3*x^3*lo g(a*x + 1)^2 - 8*a^3*x^3*log(a*x + 1)*log(a*x - 1) - 4*a^3*x^3*log(a*x - 1 )^2 + 11*a^2*x^2 - 1)/x^3)*a^2 - 1/30*(16*a^4*log(a^2*x^2 - 1) - 16*a^4*lo g(x^2) - (14*a^2*x^2 - 3)/x^4)*a*arctanh(a*x) - 1/15*(15*a^4*x^4 - 10*a^2* x^2 + 3)*arctanh(a*x)^2/x^5
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^6} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^6} \,d x \]