Integrand size = 20, antiderivative size = 67 \[ \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx=\frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{2 c} \]
arctanh(a*x)^2*ln(2-2/(-a*x+1))/c+arctanh(a*x)*polylog(2,-1+2/(-a*x+1))/c- 1/2*polylog(3,-1+2/(-a*x+1))/c
Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx=\frac {\text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )}{c}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )}{c}-\frac {\operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )}{2 c} \]
(ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])])/c + (ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])])/c - PolyLog[3, E^(2*ArcTanh[a*x])]/(2*c)
Time = 0.50 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2026, 6494, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x (c-a c x)}dx\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{c}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {2 a \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {2 a \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\) |
(ArcTanh[a*x]^2*Log[2 - 2/(1 - a*x)])/c - (2*a*(-1/2*(ArcTanh[a*x]*PolyLog [2, -1 + 2/(1 - a*x)])/a + PolyLog[3, -1 + 2/(1 - a*x)]/(4*a)))/c
3.2.19.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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[(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 = 13.16 (sec) , antiderivative size = 647, normalized size of antiderivative = 9.66
| method | result | size |
| derivativedivides | \(\frac {\frac {a \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )}{c}-\frac {a \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{c}+\frac {2 a \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\left (2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}-2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\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}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )-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 )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+2 i \pi +2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4}\right )}{c}}{a}\) | \(647\) |
| default | \(\frac {\frac {a \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x \right )}{c}-\frac {a \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{c}+\frac {2 a \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\left (2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}-2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\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}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )-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 )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+2 i \pi +2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{2}}{4}\right )}{c}}{a}\) | \(647\) |
| parts | \(\text {Expression too large to display}\) | \(1015\) |
1/a*(a/c*arctanh(a*x)^2*ln(a*x)-a/c*arctanh(a*x)^2*ln(a*x-1)+2*a/c*(-1/2*a rctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)+1/2*arctanh(a*x)^2*ln(1-(a*x+1) /(-a^2*x^2+1)^(1/2))+arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-po lylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2* x^2+1)^(1/2))+arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-polylog( 3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*(2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+ 1))^3-2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^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))*csgn(I*(-(a*x+1)^2/(a^2 *x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))-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)/(-(a*x+1)^2/(a^2*x^2-1)+1))^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)/(-(a* x+1)^2/(a^2*x^2-1)+1))^2+I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^ 2/(a^2*x^2-1)+1))^3+2*I*Pi+2*ln(2))*arctanh(a*x)^2))
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.27 \[ \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx=\frac {\log \left (\frac {2 \, a x}{a x - 1}\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2 \, {\rm Li}_2\left (-\frac {2 \, a x}{a x - 1} + 1\right ) \log \left (-\frac {a x + 1}{a x - 1}\right ) - 2 \, {\rm polylog}\left (3, -\frac {a x + 1}{a x - 1}\right )}{4 \, c} \]
1/4*(log(2*a*x/(a*x - 1))*log(-(a*x + 1)/(a*x - 1))^2 + 2*dilog(-2*a*x/(a* x - 1) + 1)*log(-(a*x + 1)/(a*x - 1)) - 2*polylog(3, -(a*x + 1)/(a*x - 1)) )/c
\[ \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx=- \frac {\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a x^{2} - x}\, dx}{c} \]
\[ \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{2}}{a c x^{2} - c x} \,d x } \]
-1/12*log(-a*x + 1)^3/c + 1/4*integrate(-(log(a*x + 1)^2 - 2*log(a*x + 1)* log(-a*x + 1))/(a*c*x^2 - c*x), x)
\[ \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{2}}{a c x^{2} - c x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{c x-a c x^2} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{c\,x-a\,c\,x^2} \,d x \]