Integrand size = 19, antiderivative size = 239 \[ \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx=\frac {a \text {arctanh}(a x)^4}{c}-\frac {\text {arctanh}(a x)^4}{c x}+\frac {a \text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {4 a \text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \text {arctanh}(a x) \operatorname {PolyLog}\left (4,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \operatorname {PolyLog}\left (5,-1+\frac {2}{1-a x}\right )}{2 c} \]
a*arctanh(a*x)^4/c-arctanh(a*x)^4/c/x+a*arctanh(a*x)^4*ln(2-2/(-a*x+1))/c+ 4*a*arctanh(a*x)^3*ln(2-2/(a*x+1))/c+2*a*arctanh(a*x)^3*polylog(2,-1+2/(-a *x+1))/c-6*a*arctanh(a*x)^2*polylog(2,-1+2/(a*x+1))/c-3*a*arctanh(a*x)^2*p olylog(3,-1+2/(-a*x+1))/c-6*a*arctanh(a*x)*polylog(3,-1+2/(a*x+1))/c+3*a*a rctanh(a*x)*polylog(4,-1+2/(-a*x+1))/c-3*a*polylog(4,-1+2/(a*x+1))/c-3/2*a *polylog(5,-1+2/(-a*x+1))/c
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.72 \[ \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx=-\frac {a \left (-\frac {\pi ^4}{16}+\frac {i \pi ^5}{160}+\text {arctanh}(a x)^4+\frac {\text {arctanh}(a x)^4}{a x}-4 \text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-\text {arctanh}(a x)^4 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-2 \text {arctanh}(a x)^2 (3+\text {arctanh}(a x)) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+3 \text {arctanh}(a x) (2+\text {arctanh}(a x)) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )-3 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )-3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (5,e^{2 \text {arctanh}(a x)}\right )\right )}{c} \]
-((a*(-1/16*Pi^4 + (I/160)*Pi^5 + ArcTanh[a*x]^4 + ArcTanh[a*x]^4/(a*x) - 4*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - ArcTanh[a*x]^4*Log[1 - E^(2 *ArcTanh[a*x])] - 2*ArcTanh[a*x]^2*(3 + ArcTanh[a*x])*PolyLog[2, E^(2*ArcT anh[a*x])] + 3*ArcTanh[a*x]*(2 + ArcTanh[a*x])*PolyLog[3, E^(2*ArcTanh[a*x ])] - 3*PolyLog[4, E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLog[4, E^(2*Ar cTanh[a*x])] + (3*PolyLog[5, E^(2*ArcTanh[a*x])])/2))/c)
Time = 2.23 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {6496, 27, 6452, 6494, 6550, 6494, 6618, 6620, 6622, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx\) |
| \(\Big \downarrow \) 6496 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^4}{x^2}dx}{c}+a \int \frac {\text {arctanh}(a x)^4}{c x (1-a x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^4}{x^2}dx}{c}+\frac {a \int \frac {\text {arctanh}(a x)^4}{x (1-a x)}dx}{c}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {4 a \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^4}{x}}{c}+\frac {a \int \frac {\text {arctanh}(a x)^4}{x (1-a x)}dx}{c}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {4 a \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^4}{x}}{c}+\frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )}{c}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle \frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )}{c}+\frac {4 a \left (\int \frac {\text {arctanh}(a x)^3}{x (a x+1)}dx+\frac {1}{4} \text {arctanh}(a x)^4\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )}{c}+\frac {4 a \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle \frac {4 a \left (-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}+\frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \int \frac {\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )}{c}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \left (\frac {3}{2} \int \frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )\right )}{c}+\frac {4 a \left (-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}\) |
| \(\Big \downarrow \) 6622 |
| \(\displaystyle \frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \left (\frac {3}{2} \int \frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )\right )}{c}+\frac {4 a \left (-3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}\) |
| \(\Big \downarrow \) 6624 |
| \(\displaystyle \frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )\right )}{c}+\frac {4 a \left (-3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}\) |
| \(\Big \downarrow \) 6624 |
| \(\displaystyle \frac {4 a \left (-3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}+\frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{2 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )\right )}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {4 a \left (-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^4}{x}}{c}+\frac {a \left (\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )-4 a \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (5,\frac {2}{1-a x}-1\right )}{4 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )\right )}{c}\) |
(-(ArcTanh[a*x]^4/x) + 4*a*(ArcTanh[a*x]^4/4 + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - 3*a*((ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*a) + (Arc Tanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/(2*a) + PolyLog[4, -1 + 2/(1 + a*x )]/(4*a))))/c + (a*(ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)] - 4*a*(-1/2*(ArcTa nh[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
3.2.39.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
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[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Simp[e/(d*f) Int[(f*x)^(m + 1)*((a + b*ArcTanh[c*x])^p/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[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[(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] & & EqQ[c^2*d + e, 0] && EqQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(572\) vs. \(2(237)=474\).
Time = 2.77 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.40
| method | result | size |
| derivativedivides | \(a \left (\frac {\operatorname {arctanh}\left (a x \right )^{4} \left (a x -1\right )}{c a x}+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \operatorname {polylog}\left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \operatorname {polylog}\left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {2 \operatorname {arctanh}\left (a x \right )^{4}}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(573\) |
| default | \(a \left (\frac {\operatorname {arctanh}\left (a x \right )^{4} \left (a x -1\right )}{c a x}+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \operatorname {polylog}\left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \operatorname {polylog}\left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {2 \operatorname {arctanh}\left (a x \right )^{4}}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(573\) |
a*(1/c*arctanh(a*x)^4/a/x*(a*x-1)+1/c*arctanh(a*x)^4*ln(1+(a*x+1)/(-a^2*x^ 2+1)^(1/2))+4/c*arctanh(a*x)^3*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-12/c *arctanh(a*x)^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+24/c*arctanh(a*x)*p olylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))-24/c*polylog(5,-(a*x+1)/(-a^2*x^2+1) ^(1/2))+1/c*arctanh(a*x)^4*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+4/c*arctanh(a* x)^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-12/c*arctanh(a*x)^2*polylog(3,( a*x+1)/(-a^2*x^2+1)^(1/2))+24/c*arctanh(a*x)*polylog(4,(a*x+1)/(-a^2*x^2+1 )^(1/2))-24/c*polylog(5,(a*x+1)/(-a^2*x^2+1)^(1/2))-2/c*arctanh(a*x)^4+4/c *arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+12/c*arctanh(a*x)^2*polyl og(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-24/c*arctanh(a*x)*polylog(3,-(a*x+1)/(-a ^2*x^2+1)^(1/2))+24/c*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+4/c*arctanh(a *x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+12/c*arctanh(a*x)^2*polylog(2,(a*x+ 1)/(-a^2*x^2+1)^(1/2))-24/c*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1 /2))+24/c*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{{\left (a c x - c\right )} x^{2}} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx=- \frac {\int \frac {\operatorname {atanh}^{4}{\left (a x \right )}}{a x^{3} - x^{2}}\, dx}{c} \]
\[ \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{{\left (a c x - c\right )} x^{2}} \,d x } \]
-1/80*(a*x*log(-a*x + 1)^5 + 5*log(-a*x + 1)^4)/(c*x) + 1/16*integrate(-(l og(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*(a*x + log(a*x + 1))*log(-a*x + 1)^3)/(a*c*x^3 - c*x^2), x)
\[ \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{{\left (a c x - c\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^4}{x^2 (c-a c x)} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^4}{x^2\,\left (c-a\,c\,x\right )} \,d x \]