Integrand size = 21, antiderivative size = 160 \[ \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx=\frac {3 b e (a+b \text {arctanh}(c+d x))^2}{2 d}+\frac {3 b e (c+d x) (a+b \text {arctanh}(c+d x))^2}{2 d}-\frac {e (a+b \text {arctanh}(c+d x))^3}{2 d}+\frac {e (c+d x)^2 (a+b \text {arctanh}(c+d x))^3}{2 d}-\frac {3 b^2 e (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^3 e \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{2 d} \]
3/2*b*e*(a+b*arctanh(d*x+c))^2/d+3/2*b*e*(d*x+c)*(a+b*arctanh(d*x+c))^2/d- 1/2*e*(a+b*arctanh(d*x+c))^3/d+1/2*e*(d*x+c)^2*(a+b*arctanh(d*x+c))^3/d-3* b^2*e*(a+b*arctanh(d*x+c))*ln(2/(-d*x-c+1))/d-3/2*b^3*e*polylog(2,(-d*x-c- 1)/(-d*x-c+1))/d
Time = 0.30 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.38 \[ \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx=\frac {e \left (6 b^2 (-1+c+d x) (b+a (1+c+d x)) \text {arctanh}(c+d x)^2+2 b^3 \left (-1+c^2+2 c d x+d^2 x^2\right ) \text {arctanh}(c+d x)^3+6 b \text {arctanh}(c+d x) \left (a (c+d x) (2 b+a c+a d x)-2 b^2 \log \left (1+e^{-2 \text {arctanh}(c+d x)}\right )\right )+a \left (6 a b c+2 a^2 c^2+6 a b d x+4 a^2 c d x+2 a^2 d^2 x^2+3 a b \log (1-c-d x)-3 a b \log (1+c+d x)-12 b^2 \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )\right )+6 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c+d x)}\right )\right )}{4 d} \]
(e*(6*b^2*(-1 + c + d*x)*(b + a*(1 + c + d*x))*ArcTanh[c + d*x]^2 + 2*b^3* (-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTanh[c + d*x]^3 + 6*b*ArcTanh[c + d*x]*( a*(c + d*x)*(2*b + a*c + a*d*x) - 2*b^2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + a*(6*a*b*c + 2*a^2*c^2 + 6*a*b*d*x + 4*a^2*c*d*x + 2*a^2*d^2*x^2 + 3*a*b *Log[1 - c - d*x] - 3*a*b*Log[1 + c + d*x] - 12*b^2*Log[1/Sqrt[1 - (c + d* x)^2]]) + 6*b^3*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/(4*d)
Time = 0.99 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6657, 27, 6452, 6542, 6436, 6510, 6546, 6470, 2849, 2752}
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.
| \(\displaystyle \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 6657 |
| \(\displaystyle \frac {\int e (c+d x) (a+b \text {arctanh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int (c+d x) (a+b \text {arctanh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \int \frac {(c+d x)^2 (a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \left (\int \frac {(a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)-\int (a+b \text {arctanh}(c+d x))^2d(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \left (2 b \int \frac {(c+d x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)+\int \frac {(a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)-\left ((c+d x) (a+b \text {arctanh}(c+d x))^2\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \left (2 b \int \frac {(c+d x) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-(c+d x) (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \left (2 b \left (\int \frac {a+b \text {arctanh}(c+d x)}{-c-d x+1}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}\right )+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-(c+d x) (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \left (2 b \left (-b \int \frac {\log \left (\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-(c+d x) (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \left (2 b \left (b \int \frac {\log \left (\frac {2}{-c-d x+1}\right )}{1-\frac {2}{-c-d x+1}}d\frac {1}{-c-d x+1}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-(c+d x) (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^3-\frac {3}{2} b \left (2 b \left (-\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))+\frac {1}{2} b \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right )\right )+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-(c+d x) (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
(e*(((c + d*x)^2*(a + b*ArcTanh[c + d*x])^3)/2 - (3*b*(-((c + d*x)*(a + b* ArcTanh[c + d*x])^2) + (a + b*ArcTanh[c + d*x])^3/(3*b) + 2*b*(-1/2*(a + b *ArcTanh[c + d*x])^2/b + (a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)] + ( b*PolyLog[2, 1 - 2/(1 - c - d*x)])/2)))/2))/d
3.1.24.3.1 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[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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_)^(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_.)/((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_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[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*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[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[(f*(x/d))^m*(a + b*ArcTanh[x])^p, x ], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(931\) vs. \(2(152)=304\).
Time = 0.65 (sec) , antiderivative size = 932, normalized size of antiderivative = 5.82
| method | result | size |
| risch | \(\frac {3 b^{3} e \operatorname {dilog}\left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2 d}-\frac {3 e d \ln \left (-d x -c +1\right ) a^{2} b \,x^{2}}{4}+\frac {3 e \ln \left (-d x -c +1\right )^{2} a \,b^{2} c x}{4}-\frac {3 e \ln \left (-d x -c +1\right ) a^{2} b c x}{2}-\frac {3 e a \,b^{2} \ln \left (-d x -c +1\right ) c}{2 d}+\frac {3 e \ln \left (-d x -c +1\right )^{2} a \,b^{2} c^{2}}{8 d}-\frac {3 e \ln \left (-d x -c +1\right ) a^{2} b \,c^{2}}{4 d}+\frac {3 e d \ln \left (-d x -c +1\right )^{2} a \,b^{2} x^{2}}{8}+\frac {3 e \,a^{2} b \ln \left (-d x -c -1\right ) c^{2}}{4 d}+\frac {3 e a \,b^{2} \ln \left (-d x -c -1\right ) c}{2 d}+e \,a^{3} c x -\frac {e d \ln \left (-d x -c +1\right )^{3} b^{3} x^{2}}{16}-\frac {e \ln \left (-d x -c +1\right )^{3} b^{3} c^{2}}{16 d}+\frac {3 e \ln \left (-d x -c +1\right ) a \,b^{2}}{2 d}-\frac {3 e a \,b^{2} \ln \left (-d x -c +1\right )^{2}}{8 d}+\frac {3 e \,a^{2} b \ln \left (-d x -c +1\right )}{4 d}+\frac {3 e \,b^{3} \ln \left (-d x -c +1\right )^{2} c}{8 d}-\frac {3 e a \,b^{2} \ln \left (-d x -c +1\right ) x}{2}-\frac {e \ln \left (-d x -c +1\right )^{3} b^{3} c x}{8}+\frac {e d \,a^{3} x^{2}}{2}+\frac {3 e \,b^{3} \ln \left (-d x -c +1\right )^{2} x}{8}-\frac {3 e \ln \left (-d x -c +1\right )^{2} b^{3}}{8 d}+\frac {e \,b^{3} \ln \left (-d x -c +1\right )^{3}}{16 d}+\frac {3 e \,a^{2} b x}{2}+\frac {e \,b^{3} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \ln \left (d x +c +1\right )^{3}}{16 d}+\frac {3 e \,b^{2} \left (-b \,d^{2} x^{2} \ln \left (-d x -c +1\right )+2 a \,d^{2} x^{2}-2 b c d x \ln \left (-d x -c +1\right )+4 a c d x -\ln \left (-d x -c +1\right ) b \,c^{2}+2 a \,c^{2}+2 b d x +2 b c +b \ln \left (-d x -c +1\right )-2 a +2 b \right ) \ln \left (d x +c +1\right )^{2}}{16 d}-\frac {e \,a^{3}}{2 d}-\frac {3 e \,a^{2} b \ln \left (-d x -c -1\right )}{4 d}+\frac {3 e a \,b^{2} \ln \left (-d x -c -1\right )}{2 d}+\frac {3 b^{3} e \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2 d}-\frac {3 b^{3} e \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (-d x -c +1\right )}{2 d}+\left (\frac {3 e \,b^{3} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \ln \left (-d x -c +1\right )^{2}}{16 d}-\frac {3 e \,b^{2} x \left (a d x +2 a c +b \right ) \ln \left (-d x -c +1\right )}{4}-\frac {3 b e \left (-a^{2} d^{2} x^{2}-2 a^{2} c d x +\ln \left (-d x -c +1\right ) a b \,c^{2}-2 a b d x +b^{2} c \ln \left (-d x -c +1\right )-\ln \left (-d x -c +1\right ) a b -\ln \left (-d x -c +1\right ) b^{2}\right )}{4 d}\right ) \ln \left (d x +c +1\right )+\frac {e \,a^{3} c^{2}}{2 d}-\frac {3 e \,a^{2} b}{2 d}+\frac {3 e \,a^{2} b c}{2 d}\) | \(932\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(5963\) |
| default | \(\text {Expression too large to display}\) | \(5963\) |
| parts | \(\text {Expression too large to display}\) | \(5970\) |
3/2*b^3*e/d*dilog(-1/2*d*x-1/2*c+1/2)-3/4*e*d*ln(-d*x-c+1)*a^2*b*x^2+3/4*e *ln(-d*x-c+1)^2*a*b^2*c*x-3/2*e*ln(-d*x-c+1)*a^2*b*c*x-3/2*e/d*a*b^2*ln(-d *x-c+1)*c+3/8*e/d*ln(-d*x-c+1)^2*a*b^2*c^2-3/4*e/d*ln(-d*x-c+1)*a^2*b*c^2+ 3/8*e*d*ln(-d*x-c+1)^2*a*b^2*x^2+3/4*e*a^2*b/d*ln(-d*x-c-1)*c^2+3/2*e*a*b^ 2/d*ln(-d*x-c-1)*c+e*a^3*c*x-1/16*e*d*ln(-d*x-c+1)^3*b^3*x^2-1/16*e/d*ln(- d*x-c+1)^3*b^3*c^2+3/2*e/d*ln(-d*x-c+1)*a*b^2-3/8*e/d*a*b^2*ln(-d*x-c+1)^2 +3/4*e/d*a^2*b*ln(-d*x-c+1)+3/8*e/d*b^3*ln(-d*x-c+1)^2*c-3/2*e*a*b^2*ln(-d *x-c+1)*x-1/8*e*ln(-d*x-c+1)^3*b^3*c*x+1/2*e*d*a^3*x^2+3/8*e*b^3*ln(-d*x-c +1)^2*x-3/8*e/d*ln(-d*x-c+1)^2*b^3+1/16*e/d*b^3*ln(-d*x-c+1)^3+3/2*e*a^2*b *x+1/16*e*b^3*(d^2*x^2+2*c*d*x+c^2-1)/d*ln(d*x+c+1)^3+3/16*e*b^2*(-b*d^2*x ^2*ln(-d*x-c+1)+2*a*d^2*x^2-2*b*c*d*x*ln(-d*x-c+1)+4*a*c*d*x-ln(-d*x-c+1)* b*c^2+2*a*c^2+2*b*d*x+2*b*c+b*ln(-d*x-c+1)-2*a+2*b)/d*ln(d*x+c+1)^2-1/2*e/ d*a^3-3/4*e*a^2*b/d*ln(-d*x-c-1)+3/2*e*a*b^2/d*ln(-d*x-c-1)+3/2*b^3*e/d*ln (-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)-3/2*b^3*e/d*ln(1/2*d*x+1/2*c+1/ 2)*ln(-d*x-c+1)+(3/16*e*b^3*(d^2*x^2+2*c*d*x+c^2-1)/d*ln(-d*x-c+1)^2-3/4*e *b^2*x*(a*d*x+2*a*c+b)*ln(-d*x-c+1)-3/4*b*e*(-a^2*d^2*x^2-2*a^2*c*d*x+ln(- d*x-c+1)*a*b*c^2-2*a*b*d*x+b^2*c*ln(-d*x-c+1)-ln(-d*x-c+1)*a*b-ln(-d*x-c+1 )*b^2)/d)*ln(d*x+c+1)+1/2*e/d*a^3*c^2-3/2*e/d*a^2*b+3/2*e/d*a^2*b*c
\[ \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
integral(a^3*d*e*x + a^3*c*e + (b^3*d*e*x + b^3*c*e)*arctanh(d*x + c)^3 + 3*(a*b^2*d*e*x + a*b^2*c*e)*arctanh(d*x + c)^2 + 3*(a^2*b*d*e*x + a^2*b*c* e)*arctanh(d*x + c), x)
\[ \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx=e \left (\int a^{3} c\, dx + \int a^{3} d x\, dx + \int b^{3} c \operatorname {atanh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c \operatorname {atanh}{\left (c + d x \right )}\, dx + \int b^{3} d x \operatorname {atanh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d x \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d x \operatorname {atanh}{\left (c + d x \right )}\, dx\right ) \]
e*(Integral(a**3*c, x) + Integral(a**3*d*x, x) + Integral(b**3*c*atanh(c + d*x)**3, x) + Integral(3*a*b**2*c*atanh(c + d*x)**2, x) + Integral(3*a**2 *b*c*atanh(c + d*x), x) + Integral(b**3*d*x*atanh(c + d*x)**3, x) + Integr al(3*a*b**2*d*x*atanh(c + d*x)**2, x) + Integral(3*a**2*b*d*x*atanh(c + d* x), x))
Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (142) = 284\).
Time = 0.40 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.93 \[ \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx=\frac {1}{2} \, a^{3} d e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a^{2} b d e + a^{3} c e x + \frac {3 \, {\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a^{2} b c e}{2 \, d} + \frac {3 \, {\left (\log \left (d x + c + 1\right ) \log \left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right )\right )} b^{3} e}{2 \, d} + \frac {3 \, {\left (c e + e\right )} a b^{2} \log \left (d x + c + 1\right )}{2 \, d} - \frac {3 \, {\left (c e - e\right )} a b^{2} \log \left (d x + c - 1\right )}{2 \, d} + \frac {24 \, a b^{2} d e x \log \left (d x + c + 1\right ) + {\left (b^{3} d^{2} e x^{2} + 2 \, b^{3} c d e x + {\left (c^{2} e - e\right )} b^{3}\right )} \log \left (d x + c + 1\right )^{3} - {\left (b^{3} d^{2} e x^{2} + 2 \, b^{3} c d e x + {\left (c^{2} e - e\right )} b^{3}\right )} \log \left (-d x - c + 1\right )^{3} + 6 \, {\left (a b^{2} d^{2} e x^{2} + {\left (c^{2} e - e\right )} a b^{2} + {\left (c e + e\right )} b^{3} + {\left (2 \, a b^{2} c d e + b^{3} d e\right )} x\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (2 \, a b^{2} d^{2} e x^{2} + 2 \, {\left (c^{2} e - e\right )} a b^{2} + 2 \, {\left (c e - e\right )} b^{3} + 2 \, {\left (2 \, a b^{2} c d e + b^{3} d e\right )} x + {\left (b^{3} d^{2} e x^{2} + 2 \, b^{3} c d e x + {\left (c^{2} e - e\right )} b^{3}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )^{2} - 3 \, {\left (8 \, a b^{2} d e x + {\left (b^{3} d^{2} e x^{2} + 2 \, b^{3} c d e x + {\left (c^{2} e - e\right )} b^{3}\right )} \log \left (d x + c + 1\right )^{2} + 4 \, {\left (a b^{2} d^{2} e x^{2} + {\left (c^{2} e - e\right )} a b^{2} + {\left (c e + e\right )} b^{3} + {\left (2 \, a b^{2} c d e + b^{3} d e\right )} x\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{16 \, d} \]
1/2*a^3*d*e*x^2 + 3/4*(2*x^2*arctanh(d*x + c) + d*(2*x/d^2 - (c^2 + 2*c + 1)*log(d*x + c + 1)/d^3 + (c^2 - 2*c + 1)*log(d*x + c - 1)/d^3))*a^2*b*d*e + a^3*c*e*x + 3/2*(2*(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))* a^2*b*c*e/d + 3/2*(log(d*x + c + 1)*log(-1/2*d*x - 1/2*c + 1/2) + dilog(1/ 2*d*x + 1/2*c + 1/2))*b^3*e/d + 3/2*(c*e + e)*a*b^2*log(d*x + c + 1)/d - 3 /2*(c*e - e)*a*b^2*log(d*x + c - 1)/d + 1/16*(24*a*b^2*d*e*x*log(d*x + c + 1) + (b^3*d^2*e*x^2 + 2*b^3*c*d*e*x + (c^2*e - e)*b^3)*log(d*x + c + 1)^3 - (b^3*d^2*e*x^2 + 2*b^3*c*d*e*x + (c^2*e - e)*b^3)*log(-d*x - c + 1)^3 + 6*(a*b^2*d^2*e*x^2 + (c^2*e - e)*a*b^2 + (c*e + e)*b^3 + (2*a*b^2*c*d*e + b^3*d*e)*x)*log(d*x + c + 1)^2 + 3*(2*a*b^2*d^2*e*x^2 + 2*(c^2*e - e)*a*b ^2 + 2*(c*e - e)*b^3 + 2*(2*a*b^2*c*d*e + b^3*d*e)*x + (b^3*d^2*e*x^2 + 2* b^3*c*d*e*x + (c^2*e - e)*b^3)*log(d*x + c + 1))*log(-d*x - c + 1)^2 - 3*( 8*a*b^2*d*e*x + (b^3*d^2*e*x^2 + 2*b^3*c*d*e*x + (c^2*e - e)*b^3)*log(d*x + c + 1)^2 + 4*(a*b^2*d^2*e*x^2 + (c^2*e - e)*a*b^2 + (c*e + e)*b^3 + (2*a *b^2*c*d*e + b^3*d*e)*x)*log(d*x + c + 1))*log(-d*x - c + 1))/d
\[ \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (c e+d e x) (a+b \text {arctanh}(c+d x))^3 \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^3 \,d x \]