Integrand size = 16, antiderivative size = 205 \[ \int \frac {x \text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {\text {arctanh}(a x)^3}{a^2 c}+\frac {x \text {arctanh}(a x)^3}{a c}-\frac {3 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2 c}+\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^2 c}-\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^2 c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )}{2 a^2 c}+\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^2 c}-\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )}{2 a^2 c}-\frac {3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+a x}\right )}{4 a^2 c} \]
arctanh(a*x)^3/a^2/c+x*arctanh(a*x)^3/a/c-3*arctanh(a*x)^2*ln(2/(-a*x+1))/ a^2/c+arctanh(a*x)^3*ln(2/(a*x+1))/a^2/c-3*arctanh(a*x)*polylog(2,1-2/(-a* x+1))/a^2/c-3/2*arctanh(a*x)^2*polylog(2,1-2/(a*x+1))/a^2/c+3/2*polylog(3, 1-2/(-a*x+1))/a^2/c-3/2*arctanh(a*x)*polylog(3,1-2/(a*x+1))/a^2/c-3/4*poly log(4,1-2/(a*x+1))/a^2/c
Time = 0.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.61 \[ \int \frac {x \text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {-\text {arctanh}(a x)^3+a x \text {arctanh}(a x)^3-3 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+\text {arctanh}(a x)^3 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-\frac {3}{2} (-2+\text {arctanh}(a x)) \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-\frac {3}{2} (-1+\text {arctanh}(a x)) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )}{a^2 c} \]
(-ArcTanh[a*x]^3 + a*x*ArcTanh[a*x]^3 - 3*ArcTanh[a*x]^2*Log[1 + E^(-2*Arc Tanh[a*x])] + ArcTanh[a*x]^3*Log[1 + E^(-2*ArcTanh[a*x])] - (3*(-2 + ArcTa nh[a*x])*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])])/2 - (3*(-1 + ArcTa nh[a*x])*PolyLog[3, -E^(-2*ArcTanh[a*x])])/2 - (3*PolyLog[4, -E^(-2*ArcTan h[a*x])])/4)/(a^2*c)
Time = 1.81 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6492, 27, 6436, 6470, 6546, 6470, 6618, 6620, 6622, 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.
| \(\displaystyle \int \frac {x \text {arctanh}(a x)^3}{a c x+c} \, dx\) |
| \(\Big \downarrow \) 6492 |
| \(\displaystyle \frac {\int \text {arctanh}(a x)^3dx}{a c}-\frac {\int \frac {\text {arctanh}(a x)^3}{c (a x+1)}dx}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \text {arctanh}(a x)^3dx}{a c}-\frac {\int \frac {\text {arctanh}(a x)^3}{a x+1}dx}{a c}\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a c}-\frac {\int \frac {\text {arctanh}(a x)^3}{a x+1}dx}{a c}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a c}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a c}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a c}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a c}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a c}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a c}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a c}-\frac {3 \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a c}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a c}-\frac {3 \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a c}\) |
| \(\Big \downarrow \) 6622 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a c}-\frac {3 \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{2 a}\right )-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a c}-\frac {3 \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,1-\frac {2}{a x+1}\right )}{4 a}\right )-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a c}\) |
(x*ArcTanh[a*x]^3 - 3*a*(-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 + Pol yLog[3, 1 - 2/(1 - a*x)]/(4*a)))/a))/(a*c) - (-((ArcTanh[a*x]^3*Log[2/(1 + a*x)])/a) + 3*((ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 + a*x)])/(2*a) + (ArcT anh[a*x]*PolyLog[3, 1 - 2/(1 + a*x)])/(2*a) + PolyLog[4, 1 - 2/(1 + a*x)]/ (4*a)))/(a*c)
3.2.28.3.1 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_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[f/e Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x]) ^p, x], x] - Simp[d*(f/e) 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] && GtQ[m, 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[(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[(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 = 12.95 (sec) , antiderivative size = 736, normalized size of antiderivative = 3.59
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{3} a x}{c}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{c}-\frac {3 \left (-\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {\operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3}-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+\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 {arctanh}\left (a x \right )^{4}}{6}+\frac {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 (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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6}+\frac {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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {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} \operatorname {arctanh}\left (a x \right )^{3}}{3}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {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 ) \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {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 (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {\ln \left (2\right ) \operatorname {arctanh}\left (a x \right )^{3}}{3}\right )}{c}}{a^{2}}\) | \(736\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{3} a x}{c}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{c}-\frac {3 \left (-\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}-\frac {\operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3}-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+\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 {arctanh}\left (a x \right )^{4}}{6}+\frac {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 (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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6}+\frac {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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {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} \operatorname {arctanh}\left (a x \right )^{3}}{3}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {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 ) \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {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 (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6}-\frac {\ln \left (2\right ) \operatorname {arctanh}\left (a x \right )^{3}}{3}\right )}{c}}{a^{2}}\) | \(736\) |
| parts | \(\frac {x \operatorname {arctanh}\left (a x \right )^{3}}{a c}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (a x +1\right )}{c \,a^{2}}-\frac {3 a \left (-\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{3}}-\frac {\operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3}}+\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3}}-\frac {\operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a^{3}}+\frac {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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{3}}-\frac {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 ) \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{3}}+\frac {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 (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 (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{3}}-\frac {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 (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{3}}-\frac {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} \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{3}}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{3}}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}{6 a^{3}}-\frac {\ln \left (2\right ) \operatorname {arctanh}\left (a x \right )^{3}}{3 a^{3}}+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{6 a^{3}}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{a^{3}}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{3 a^{3}}+\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^{3}}-\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3}}\right )}{c}\) | \(789\) |
1/a^2*(1/c*arctanh(a*x)^3*a*x-1/c*arctanh(a*x)^3*ln(a*x+1)-3/c*(-2/3*arcta nh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*arctanh(a*x)^2*polylog(2,-(a* x+1)^2/(-a^2*x^2+1))+1/2*arctanh(a*x)*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))-1 /4*polylog(4,-(a*x+1)^2/(-a^2*x^2+1))-1/3*arctanh(a*x)^3-1/2*polylog(3,-(a *x+1)^2/(-a^2*x^2+1))+arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)+1)+arctanh( a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+1/6*arctanh(a*x)^4+1/6*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))*csgn(I*(a*x+1)^ 2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))*arctanh(a*x)^3+1/6*I*Pi*csgn(I*( a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1) +1))^2*arctanh(a*x)^3-1/3*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*( a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^3-1/6*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2- 1))^3*arctanh(a*x)^3-1/6*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^ 2*x^2-1)+1))^3*arctanh(a*x)^3-1/6*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))*arctanh(a*x)^3-1/6*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)/(-(a*x+1)^2/(a^2*x^2-1)+1))^ 2*arctanh(a*x)^3-1/3*ln(2)*arctanh(a*x)^3))
\[ \int \frac {x \text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {x \operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \]
\[ \int \frac {x \text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {\int \frac {x \operatorname {atanh}^{3}{\left (a x \right )}}{a x + 1}\, dx}{c} \]
\[ \int \frac {x \text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {x \operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \]
-1/8*(a*x - log(a*x + 1))*log(-a*x + 1)^3/(a^2*c) + 1/8*integrate(((a^2*x^ 2 - a*x)*log(a*x + 1)^3 - 3*(a^2*x^2 - a*x)*log(a*x + 1)^2*log(-a*x + 1) + 3*(a^2*x^2 + a*x + (a^2*x^2 - 2*a*x - 1)*log(a*x + 1))*log(-a*x + 1)^2)/( a^3*c*x^2 - a*c), x)
\[ \int \frac {x \text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {x \operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \]
Timed out. \[ \int \frac {x \text {arctanh}(a x)^3}{c+a c x} \, dx=\int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^3}{c+a\,c\,x} \,d x \]