Integrand size = 20, antiderivative size = 78 \[ \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\frac {\text {arctanh}(a x)^3}{3 a^2}+\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^2}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^2} \]
-1/3*arctanh(a*x)^3/a^2+arctanh(a*x)^2*ln(2/(-a*x+1))/a^2+arctanh(a*x)*pol ylog(2,1-2/(-a*x+1))/a^2-1/2*polylog(3,1-2/(-a*x+1))/a^2
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\frac {-\frac {1}{3} \text {arctanh}(a x)^3-\text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{a^2} \]
-((-1/3*ArcTanh[a*x]^3 - ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + Arc Tanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + PolyLog[3, -E^(-2*ArcTanh[a*x ])]/2)/a^2)
Time = 0.54 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6546, 6470, 6620, 7164}
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.
| \(\displaystyle \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\) |
-1/3*ArcTanh[a*x]^3/a^2 + ((ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a - 2*(-1/2*( ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/ (4*a)))/a
3.3.36.3.1 Defintions of rubi rules used
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_.)*(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]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.29 (sec) , antiderivative size = 638, normalized size of antiderivative = 8.18
| method | result | size |
| derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3}+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3}+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}-2 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2}+2 i \pi +i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )+4 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4}}{a^{2}}\) | \(638\) |
| default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3}+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{3}+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}-2 i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )}^{2}+2 i \pi +i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}}\right )+4 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4}}{a^{2}}\) | \(638\) |
| parts | \(-\frac {\ln \left (a^{2} x^{2}-1\right ) \operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2}}+\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3 a}+\frac {\left (-2 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (i \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )\right ) {\operatorname {csgn}\left (i \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}\right )}^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}\right )^{2}+2 i \pi +i \pi {\operatorname {csgn}\left (i \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}\right )-i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+4 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4 a}+\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a}}{a}\) | \(785\) |
1/a^2*(-1/2*arctanh(a*x)^2*ln(a*x-1)-1/2*arctanh(a*x)^2*ln(a*x+1)+arctanh( a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/3*arctanh(a*x)^3+arctanh(a*x)*poly log(2,-(a*x+1)^2/(-a^2*x^2+1))-1/2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))+1/4* (-I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-( a*x+1)^2/(a^2*x^2-1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))+I*Pi*csgn(I*(a*x+1)/( -a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))+I*Pi*csgn(I*(a*x+1)^2/( a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)) ^3-I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I*( a*x+1)^2/(a^2*x^2-1))+2*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^3+2*I*Pi*cs gn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2-2*I*Pi*cs gn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^2+2*I*Pi+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1) /(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))+4*ln(2))*a rctanh(a*x)^2)
\[ \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=\int { -\frac {x \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
\[ \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=- \int \frac {x \operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \]
\[ \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=\int { -\frac {x \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
-1/24*(3*log(a*x + 1)*log(-a*x + 1)^2 + log(-a*x + 1)^3)/a^2 + 1/4*integra te(-(a*x*log(a*x + 1)^2 - (3*a*x + 1)*log(a*x + 1)*log(-a*x + 1))/(a^3*x^2 - a), x)
\[ \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=\int { -\frac {x \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^2}{a^2\,x^2-1} \,d x \]