Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=-\frac {a \text {arctanh}(a x)}{x}+a^3 x \text {arctanh}(a x)-\frac {\text {arctanh}(a x)^2}{2 x^2}+\frac {1}{2} a^4 x^2 \text {arctanh}(a x)^2-4 a^2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )+a^2 \log (x)+2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )-2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )-a^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )+a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \]
-a*arctanh(a*x)/x+a^3*x*arctanh(a*x)-1/2*arctanh(a*x)^2/x^2+1/2*a^4*x^2*ar ctanh(a*x)^2+4*a^2*arctanh(a*x)^2*arctanh(-1+2/(-a*x+1))+a^2*ln(x)+2*a^2*a rctanh(a*x)*polylog(2,1-2/(-a*x+1))-2*a^2*arctanh(a*x)*polylog(2,-1+2/(-a* x+1))-a^2*polylog(3,1-2/(-a*x+1))+a^2*polylog(3,-1+2/(-a*x+1))
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.23 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=-\frac {a \text {arctanh}(a x)}{x}+a^3 x \text {arctanh}(a x)+\frac {1}{2} a^2 \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2}{2 x^2}+\frac {4}{3} a^2 \text {arctanh}(a x)^3+2 a^2 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-2 a^2 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+a^2 \log (x)-2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-a^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+a^2 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right ) \]
-((a*ArcTanh[a*x])/x) + a^3*x*ArcTanh[a*x] + (a^2*(-1 + a^2*x^2)*ArcTanh[a *x]^2)/2 + ((-1 + a^2*x^2)*ArcTanh[a*x]^2)/(2*x^2) + (4*a^2*ArcTanh[a*x]^3 )/3 + 2*a^2*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - 2*a^2*ArcTanh[a* x]^2*Log[1 - E^(2*ArcTanh[a*x])] + a^2*Log[x] - 2*a^2*ArcTanh[a*x]*PolyLog [2, -E^(-2*ArcTanh[a*x])] - 2*a^2*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x ])] - a^2*PolyLog[3, -E^(-2*ArcTanh[a*x])] + a^2*PolyLog[3, E^(2*ArcTanh[a *x])]
Time = 0.70 (sec) , antiderivative size = 162, 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.
| \(\displaystyle \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \int \left (a^4 x \text {arctanh}(a x)^2-\frac {2 a^2 \text {arctanh}(a x)^2}{x}+\frac {\text {arctanh}(a x)^2}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} a^4 x^2 \text {arctanh}(a x)^2+a^3 x \text {arctanh}(a x)+2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )-2 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )-4 a^2 \text {arctanh}(a x)^2 \text {arctanh}\left (1-\frac {2}{1-a x}\right )-a^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )+a^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )+a^2 \log (x)-\frac {\text {arctanh}(a x)^2}{2 x^2}-\frac {a \text {arctanh}(a x)}{x}\) |
-((a*ArcTanh[a*x])/x) + a^3*x*ArcTanh[a*x] - ArcTanh[a*x]^2/(2*x^2) + (a^4 *x^2*ArcTanh[a*x]^2)/2 - 4*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] + a ^2*Log[x] + 2*a^2*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] - 2*a^2*ArcTanh [a*x]*PolyLog[2, -1 + 2/(1 - a*x)] - a^2*PolyLog[3, 1 - 2/(1 - a*x)] + a^2 *PolyLog[3, -1 + 2/(1 - a*x)]
3.3.10.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.19 (sec) , antiderivative size = 779, normalized size of antiderivative = 4.81
| method | result | size |
| derivativedivides | \(a^{2} \left (\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+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 )+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )^{2}-i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}+\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )-\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\left (a x -\sqrt {-a^{2} x^{2}+1}+1\right ) \operatorname {arctanh}\left (a x \right )}{2 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +\sqrt {-a^{2} x^{2}+1}+1\right )}{2 a x}+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )\right )\) | \(779\) |
| default | \(a^{2} \left (\frac {a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2}-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+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 )+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {arctanh}\left (a x \right )^{2}-i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}+\left (a x +1\right ) \operatorname {arctanh}\left (a x \right )-\ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\left (a x -\sqrt {-a^{2} x^{2}+1}+1\right ) \operatorname {arctanh}\left (a x \right )}{2 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \left (a x +\sqrt {-a^{2} x^{2}+1}+1\right )}{2 a x}+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )\right )\) | \(779\) |
| parts | \(\text {Expression too large to display}\) | \(1178\) |
a^2*(1/2*a^2*x^2*arctanh(a*x)^2-2*arctanh(a*x)^2*ln(a*x)-1/2*arctanh(a*x)^ 2/a^2/x^2+2*arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)-2*arctanh(a*x)^2*l n(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-4*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2 +1)^(1/2))+4*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a*x)^2*ln(1+( a*x+1)/(-a^2*x^2+1)^(1/2))-4*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^ (1/2))+4*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,- (a*x+1)^2/(-a^2*x^2+1))-polylog(3,-(a*x+1)^2/(-a^2*x^2+1))+I*Pi*csgn(I*(-( a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/( a^2*x^2-1)))*arctanh(a*x)^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)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*arctanh(a*x)^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)-1)/(1 -(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*arctanh(a*x)^2- I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3*arctan h(a*x)^2+(a*x+1)*arctanh(a*x)-ln(1+(a*x+1)^2/(-a^2*x^2+1))-1/2*(a*x-(-a^2* x^2+1)^(1/2)+1)/a/x*arctanh(a*x)-1/2*arctanh(a*x)*(a*x+(-a^2*x^2+1)^(1/2)+ 1)/a/x+ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1))
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=\int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{3}} \,d x } \]
-1/16*(2*x^2*log(-a*x + 1) - a*((a*x^2 + 2*x)/a^2 + 2*log(a*x - 1)/a^3))*a ^4 - 1/2*a^4*integrate(x*log(a*x + 1)*log(-a*x + 1), x) + 1/4*a^3*integrat e(a*x*log(a*x + 1)^2, x) + 1/4*a^3*integrate(log(a*x + 1)^2/(a^3*x^3), x) + 1/4*(a*x - (a*x - 1)*log(-a*x + 1) - 1)*a^2 - 1/2*a^2*integrate(log(a*x + 1)^2/x, x) + a^2*integrate(log(a*x + 1)*log(-a*x + 1)/x, x) - 1/4*a^2*in tegrate(log(-a*x + 1)/x, x) - 1/4*(a*(log(a*x - 1) - log(x)) - log(-a*x + 1)/x)*a + 1/8*(a^4*x^4 - 1)*log(-a*x + 1)^2/x^2 - 1/2*integrate(log(a*x + 1)*log(-a*x + 1)/x^3, x)
\[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^3} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^3} \,d x \]