Integrand size = 19, antiderivative size = 118 \[ \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx=\frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \operatorname {PolyLog}\left (5,-1+\frac {2}{1-a x}\right )}{2 c} \]
arctanh(a*x)^4*ln(2-2/(-a*x+1))/c+2*arctanh(a*x)^3*polylog(2,-1+2/(-a*x+1) )/c-3*arctanh(a*x)^2*polylog(3,-1+2/(-a*x+1))/c+3*arctanh(a*x)*polylog(4,- 1+2/(-a*x+1))/c-3/2*polylog(5,-1+2/(-a*x+1))/c
Time = 0.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx=\frac {\text {arctanh}(a x)^4 \log \left (1-e^{2 \text {arctanh}(a x)}\right )}{c}+\frac {2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )}{c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )}{c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )}{c}-\frac {3 \operatorname {PolyLog}\left (5,e^{2 \text {arctanh}(a x)}\right )}{2 c} \]
(ArcTanh[a*x]^4*Log[1 - E^(2*ArcTanh[a*x])])/c + (2*ArcTanh[a*x]^3*PolyLog [2, E^(2*ArcTanh[a*x])])/c - (3*ArcTanh[a*x]^2*PolyLog[3, E^(2*ArcTanh[a*x ])])/c + (3*ArcTanh[a*x]*PolyLog[4, E^(2*ArcTanh[a*x])])/c - (3*PolyLog[5, E^(2*ArcTanh[a*x])])/(2*c)
Time = 0.78 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6494, 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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \int \frac {\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{c}\) |
\(\Big \downarrow \) 6620 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \int \frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\) |
\(\Big \downarrow \) 6624 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\) |
\(\Big \downarrow \) 6624 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{2 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (5,\frac {2}{1-a x}-1\right )}{4 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\) |
(ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c - (4*a*(-1/2*(ArcTanh[a*x]^3*PolyL og[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
3.2.37.3.1 Defintions of rubi rules used
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[(((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 = 1.79 (sec) , antiderivative size = 754, normalized size of antiderivative = 6.39
method | result | size |
derivativedivides | \(\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x \right )}{c}-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {-\operatorname {arctanh}\left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \operatorname {polylog}\left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \operatorname {polylog}\left (5, -\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 )^{4}}{2}}{c}\) | \(754\) |
default | \(\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x \right )}{c}-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {-\operatorname {arctanh}\left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \operatorname {polylog}\left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \operatorname {polylog}\left (5, -\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 )^{4}}{2}}{c}\) | \(754\) |
parts | \(\text {Expression too large to display}\) | \(1141\) |
1/c*arctanh(a*x)^4*ln(a*x)-1/c*arctanh(a*x)^4*ln(a*x-1)+4/c*(-1/4*arctanh( a*x)^4*ln((a*x+1)^2/(-a^2*x^2+1)-1)+1/4*arctanh(a*x)^4*ln(1-(a*x+1)/(-a^2* x^2+1)^(1/2))+arctanh(a*x)^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*arcta nh(a*x)^2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(4,( a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(5,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*arc tanh(a*x)^4*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^3*polylog(2,-(a* x+1)/(-a^2*x^2+1)^(1/2))-3*arctanh(a*x)^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^ (1/2))+6*arctanh(a*x)*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(5,- (a*x+1)/(-a^2*x^2+1)^(1/2))+1/8*(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*csg n(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)^4)
Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.31 \[ \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx=\frac {\log \left (\frac {2 \, a x}{a x - 1}\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 4 \, {\rm Li}_2\left (-\frac {2 \, a x}{a x - 1} + 1\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 12 \, \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} {\rm polylog}\left (3, -\frac {a x + 1}{a x - 1}\right ) + 24 \, \log \left (-\frac {a x + 1}{a x - 1}\right ) {\rm polylog}\left (4, -\frac {a x + 1}{a x - 1}\right ) - 24 \, {\rm polylog}\left (5, -\frac {a x + 1}{a x - 1}\right )}{16 \, c} \]
1/16*(log(2*a*x/(a*x - 1))*log(-(a*x + 1)/(a*x - 1))^4 + 4*dilog(-2*a*x/(a *x - 1) + 1)*log(-(a*x + 1)/(a*x - 1))^3 - 12*log(-(a*x + 1)/(a*x - 1))^2* polylog(3, -(a*x + 1)/(a*x - 1)) + 24*log(-(a*x + 1)/(a*x - 1))*polylog(4, -(a*x + 1)/(a*x - 1)) - 24*polylog(5, -(a*x + 1)/(a*x - 1)))/c
\[ \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx=- \frac {\int \frac {\operatorname {atanh}^{4}{\left (a x \right )}}{a x^{2} - x}\, dx}{c} \]
\[ \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{{\left (a c x - c\right )} x} \,d x } \]
-1/80*log(-a*x + 1)^5/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)*log(- a*x + 1)^3)/(a*c*x^2 - c*x), x)
\[ \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{{\left (a c x - c\right )} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^4}{x\,\left (c-a\,c\,x\right )} \,d x \]