Integrand size = 16, antiderivative size = 131 \[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}+\frac {2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a c}-\frac {3 \operatorname {PolyLog}\left (5,1-\frac {2}{1-a x}\right )}{2 a c} \]
arctanh(a*x)^4*ln(2/(-a*x+1))/a/c+2*arctanh(a*x)^3*polylog(2,1-2/(-a*x+1)) /a/c-3*arctanh(a*x)^2*polylog(3,1-2/(-a*x+1))/a/c+3*arctanh(a*x)*polylog(4 ,1-2/(-a*x+1))/a/c-3/2*polylog(5,1-2/(-a*x+1))/a/c
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=-\frac {-\frac {2}{5} \text {arctanh}(a x)^5-\text {arctanh}(a x)^4 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (5,-e^{-2 \text {arctanh}(a x)}\right )}{a c} \]
-(((-2*ArcTanh[a*x]^5)/5 - ArcTanh[a*x]^4*Log[1 + E^(-2*ArcTanh[a*x])] + 2 *ArcTanh[a*x]^3*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]^2*PolyLo g[3, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*PolyLog[4, -E^(-2*ArcTanh[a*x] )] + (3*PolyLog[5, -E^(-2*ArcTanh[a*x])])/2)/(a*c))
Time = 0.76 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6470, 6620, 6624, 6624, 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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6470 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \int \frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{c}\) |
\(\Big \downarrow \) 6620 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \left (\frac {3}{2} \int \frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{c}\) |
\(\Big \downarrow \) 6624 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{c}\) |
\(\Big \downarrow \) 6624 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{2 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{c}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (5,1-\frac {2}{1-a x}\right )}{4 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{c}\) |
(ArcTanh[a*x]^4*Log[2/(1 - a*x)])/(a*c) - (4*(-1/2*(ArcTanh[a*x]^3*PolyLog [2, 1 - 2/(1 - a*x)])/a + (3*((ArcTanh[a*x]^2*PolyLog[3, 1 - 2/(1 - a*x)]) /(2*a) - (ArcTanh[a*x]*PolyLog[4, 1 - 2/(1 - a*x)])/(2*a) + PolyLog[5, 1 - 2/(1 - a*x)]/(4*a)))/2))/c
3.2.36.3.1 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[(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[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[k + 1, u]/(2* c*d)), x] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && E qQ[c^2*d + e, 0] && EqQ[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 = 1.67 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.74
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {\left (i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}+2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}}{c}}{a}\) | \(228\) |
default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {\left (i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}+2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}}{c}}{a}\) | \(228\) |
parts | \(-\frac {\ln \left (a x -1\right ) \operatorname {arctanh}\left (a x \right )^{4}}{a c}+\frac {\frac {\left (i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}}{a}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a}}{c}\) | \(242\) |
1/a*(-1/c*arctanh(a*x)^4*ln(a*x-1)+4/c*(1/4*(I*Pi*csgn(I/(-(a*x+1)^2/(a^2* x^2-1)+1))^3-I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+I*Pi+ln(2))*arctanh (a*x)^4+1/2*arctanh(a*x)^3*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-3/4*arctanh( a*x)^2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))+3/4*arctanh(a*x)*polylog(4,-(a*x +1)^2/(-a^2*x^2+1))-3/8*polylog(5,-(a*x+1)^2/(-a^2*x^2+1))))
\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x - c} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=- \frac {\int \frac {\operatorname {atanh}^{4}{\left (a x \right )}}{a x - 1}\, dx}{c} \]
\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x - c} \,d x } \]
-1/80*log(-a*x + 1)^5/(a*c) + 1/16*integrate(-(log(a*x + 1)^4 - 4*log(a*x + 1)^3*log(-a*x + 1) + 6*log(a*x + 1)^2*log(-a*x + 1)^2 - 4*log(a*x + 1)*l og(-a*x + 1)^3)/(a*c*x - c), x)
\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x - c} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^4}{c-a\,c\,x} \,d x \]