Integrand size = 20, antiderivative size = 77 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx=-\frac {a}{12 x^3}+\frac {3 a^3}{4 x}-\frac {3}{4} a^4 \text {arctanh}(a x)-\frac {\text {arctanh}(a x)}{4 x^4}+\frac {a^2 \text {arctanh}(a x)}{x^2}-\frac {1}{2} a^4 \operatorname {PolyLog}(2,-a x)+\frac {1}{2} a^4 \operatorname {PolyLog}(2,a x) \]
-1/12*a/x^3+3/4*a^3/x-3/4*a^4*arctanh(a*x)-1/4*arctanh(a*x)/x^4+a^2*arctan h(a*x)/x^2-1/2*a^4*polylog(2,-a*x)+1/2*a^4*polylog(2,a*x)
Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.16 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx=-\frac {a}{12 x^3}+\frac {3 a^3}{4 x}-\frac {\text {arctanh}(a x)}{4 x^4}+\frac {a^2 \text {arctanh}(a x)}{x^2}+\frac {3}{8} a^4 \log (1-a x)-\frac {3}{8} a^4 \log (1+a x)+\frac {1}{2} a^4 (-\operatorname {PolyLog}(2,-a x)+\operatorname {PolyLog}(2,a x)) \]
-1/12*a/x^3 + (3*a^3)/(4*x) - ArcTanh[a*x]/(4*x^4) + (a^2*ArcTanh[a*x])/x^ 2 + (3*a^4*Log[1 - a*x])/8 - (3*a^4*Log[1 + a*x])/8 + (a^4*(-PolyLog[2, -( a*x)] + PolyLog[2, a*x]))/2
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \int \left (\frac {a^4 \text {arctanh}(a x)}{x}-\frac {2 a^2 \text {arctanh}(a x)}{x^3}+\frac {\text {arctanh}(a x)}{x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{4} a^4 \text {arctanh}(a x)-\frac {1}{2} a^4 \operatorname {PolyLog}(2,-a x)+\frac {1}{2} a^4 \operatorname {PolyLog}(2,a x)+\frac {3 a^3}{4 x}+\frac {a^2 \text {arctanh}(a x)}{x^2}-\frac {\text {arctanh}(a x)}{4 x^4}-\frac {a}{12 x^3}\) |
-1/12*a/x^3 + (3*a^3)/(4*x) - (3*a^4*ArcTanh[a*x])/4 - ArcTanh[a*x]/(4*x^4 ) + (a^2*ArcTanh[a*x])/x^2 - (a^4*PolyLog[2, -(a*x)])/2 + (a^4*PolyLog[2, a*x])/2
3.3.1.3.1 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 = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(a^{4} \left (-\frac {\operatorname {arctanh}\left (a x \right )}{4 a^{4} x^{4}}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )+\frac {\operatorname {arctanh}\left (a x \right )}{a^{2} x^{2}}-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {1}{12 a^{3} x^{3}}+\frac {3}{4 a x}+\frac {3 \ln \left (a x -1\right )}{8}-\frac {3 \ln \left (a x +1\right )}{8}\right )\) | \(96\) |
| default | \(a^{4} \left (-\frac {\operatorname {arctanh}\left (a x \right )}{4 a^{4} x^{4}}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )+\frac {\operatorname {arctanh}\left (a x \right )}{a^{2} x^{2}}-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {1}{12 a^{3} x^{3}}+\frac {3}{4 a x}+\frac {3 \ln \left (a x -1\right )}{8}-\frac {3 \ln \left (a x +1\right )}{8}\right )\) | \(96\) |
| parts | \(\frac {a^{2} \operatorname {arctanh}\left (a x \right )}{x^{2}}+\operatorname {arctanh}\left (a x \right ) a^{4} \ln \left (x \right )-\frac {\operatorname {arctanh}\left (a x \right )}{4 x^{4}}-\frac {a \left (-4 a^{4} \left (\frac {\left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{2 a}-\frac {\operatorname {dilog}\left (a x \right )}{2 a}-\frac {\operatorname {dilog}\left (a x +1\right )}{2 a}-\frac {\ln \left (x \right ) \ln \left (a x +1\right )}{2 a}\right )-\frac {3 a^{3} \ln \left (a x -1\right )}{2}+\frac {3 a^{3} \ln \left (a x +1\right )}{2}+\frac {1}{3 x^{3}}-\frac {3 a^{2}}{x}\right )}{4}\) | \(131\) |
| risch | \(\frac {3 a^{4} \ln \left (a x \right )}{8}+\frac {3 a^{3}}{4 x}-\frac {3 a^{4} \ln \left (a x +1\right )}{8}+\frac {a^{2} \ln \left (a x +1\right )}{2 x^{2}}-\frac {a^{4} \operatorname {dilog}\left (a x +1\right )}{2}-\frac {a}{12 x^{3}}-\frac {\ln \left (a x +1\right )}{8 x^{4}}-\frac {3 a^{4} \ln \left (-a x \right )}{8}+\frac {3 a^{4} \ln \left (-a x +1\right )}{8}-\frac {a^{2} \ln \left (-a x +1\right )}{2 x^{2}}+\frac {a^{4} \operatorname {dilog}\left (-a x +1\right )}{2}+\frac {\ln \left (-a x +1\right )}{8 x^{4}}\) | \(133\) |
| meijerg | \(-\frac {i a^{4} \left (-\frac {i}{3 x^{3} a^{3}}-\frac {i}{x a}+\frac {4 i \left (\frac {3}{8}-\frac {3 a^{4} x^{4}}{8}\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 x^{3} a^{3} \sqrt {a^{2} x^{2}}}\right )}{4}-\frac {i a^{4} \left (\frac {2 i a x \operatorname {polylog}\left (2, \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-\frac {2 i a x \operatorname {polylog}\left (2, -\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}-\frac {i a^{4} \left (\frac {2 i}{x a}+\frac {2 i \left (-a x +1\right ) \left (a x +1\right ) \operatorname {arctanh}\left (a x \right )}{x^{2} a^{2}}\right )}{2}\) | \(183\) |
a^4*(-1/4*arctanh(a*x)/a^4/x^4+arctanh(a*x)*ln(a*x)+arctanh(a*x)/a^2/x^2-1 /2*dilog(a*x)-1/2*dilog(a*x+1)-1/2*ln(a*x)*ln(a*x+1)-1/12/a^3/x^3+3/4/a/x+ 3/8*ln(a*x-1)-3/8*ln(a*x+1))
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x^{5}} \,d x } \]
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx=\int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}{\left (a x \right )}}{x^{5}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.45 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx=-\frac {1}{24} \, {\left (12 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a^{3} - 12 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a^{3} + 9 \, a^{3} \log \left (a x + 1\right ) - 9 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (9 \, a^{2} x^{2} - 1\right )}}{x^{3}}\right )} a + \frac {1}{4} \, {\left (2 \, a^{4} \log \left (x^{2}\right ) + \frac {4 \, a^{2} x^{2} - 1}{x^{4}}\right )} \operatorname {artanh}\left (a x\right ) \]
-1/24*(12*(log(a*x + 1)*log(x) + dilog(-a*x))*a^3 - 12*(log(-a*x + 1)*log( x) + dilog(a*x))*a^3 + 9*a^3*log(a*x + 1) - 9*a^3*log(a*x - 1) - 2*(9*a^2* x^2 - 1)/x^3)*a + 1/4*(2*a^4*log(x^2) + (4*a^2*x^2 - 1)/x^4)*arctanh(a*x)
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^5} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2}{x^5} \,d x \]