Integrand size = 23, antiderivative size = 263 \[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx=a b^2 e^2 x+\frac {b^3 e^2 (c+d x) \text {arctanh}(c+d x)}{d}-\frac {b e^2 (a+b \text {arctanh}(c+d x))^2}{2 d}+\frac {b e^2 (c+d x)^2 (a+b \text {arctanh}(c+d x))^2}{2 d}+\frac {e^2 (a+b \text {arctanh}(c+d x))^3}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arctanh}(c+d x))^3}{3 d}-\frac {b e^2 (a+b \text {arctanh}(c+d x))^2 \log \left (\frac {2}{1-c-d x}\right )}{d}+\frac {b^3 e^2 \log \left (1-(c+d x)^2\right )}{2 d}-\frac {b^2 e^2 (a+b \text {arctanh}(c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {b^3 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d} \]
a*b^2*e^2*x+b^3*e^2*(d*x+c)*arctanh(d*x+c)/d-1/2*b*e^2*(a+b*arctanh(d*x+c) )^2/d+1/2*b*e^2*(d*x+c)^2*(a+b*arctanh(d*x+c))^2/d+1/3*e^2*(a+b*arctanh(d* x+c))^3/d+1/3*e^2*(d*x+c)^3*(a+b*arctanh(d*x+c))^3/d-b*e^2*(a+b*arctanh(d* x+c))^2*ln(2/(-d*x-c+1))/d+1/2*b^3*e^2*ln(1-(d*x+c)^2)/d-b^2*e^2*(a+b*arct anh(d*x+c))*polylog(2,1-2/(-d*x-c+1))/d+1/2*b^3*e^2*polylog(3,1-2/(-d*x-c+ 1))/d
Time = 0.55 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.28 \[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx=\frac {e^2 \left (3 a^2 b (c+d x)^2+2 a^3 (c+d x)^3+6 a^2 b (c+d x)^3 \text {arctanh}(c+d x)+3 a^2 b \log \left (1-(c+d x)^2\right )+6 a b^2 \left (c+d x-\text {arctanh}(c+d x)+(c+d x)^2 \text {arctanh}(c+d x)-\text {arctanh}(c+d x)^2+(c+d x)^3 \text {arctanh}(c+d x)^2-2 \text {arctanh}(c+d x) \log \left (1+e^{-2 \text {arctanh}(c+d x)}\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c+d x)}\right )\right )+b^3 \left (6 (c+d x) \text {arctanh}(c+d x)-3 \left (1-(c+d x)^2\right ) \text {arctanh}(c+d x)^2-2 \text {arctanh}(c+d x)^3+2 (c+d x) \text {arctanh}(c+d x)^3-2 (c+d x) \left (1-(c+d x)^2\right ) \text {arctanh}(c+d x)^3-6 \text {arctanh}(c+d x)^2 \log \left (1+e^{-2 \text {arctanh}(c+d x)}\right )-6 \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+6 \text {arctanh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c+d x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c+d x)}\right )\right )\right )}{6 d} \]
(e^2*(3*a^2*b*(c + d*x)^2 + 2*a^3*(c + d*x)^3 + 6*a^2*b*(c + d*x)^3*ArcTan h[c + d*x] + 3*a^2*b*Log[1 - (c + d*x)^2] + 6*a*b^2*(c + d*x - ArcTanh[c + d*x] + (c + d*x)^2*ArcTanh[c + d*x] - ArcTanh[c + d*x]^2 + (c + d*x)^3*Ar cTanh[c + d*x]^2 - 2*ArcTanh[c + d*x]*Log[1 + E^(-2*ArcTanh[c + d*x])] + P olyLog[2, -E^(-2*ArcTanh[c + d*x])]) + b^3*(6*(c + d*x)*ArcTanh[c + d*x] - 3*(1 - (c + d*x)^2)*ArcTanh[c + d*x]^2 - 2*ArcTanh[c + d*x]^3 + 2*(c + d* x)*ArcTanh[c + d*x]^3 - 2*(c + d*x)*(1 - (c + d*x)^2)*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*Log[1 + E^(-2*ArcTanh[c + d*x])] - 6*Log[1/Sqrt[1 - (c + d*x)^2]] + 6*ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTanh[c + d*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])))/(6*d)
Time = 1.58 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.84, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6657, 27, 6452, 6542, 6452, 6542, 2009, 6510, 6546, 6470, 6620, 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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 6657 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arctanh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arctanh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \int \frac {(c+d x)^3 (a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (\int \frac {(c+d x) (a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)-\int (c+d x) (a+b \text {arctanh}(c+d x))^2d(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (b \int \frac {(c+d x)^2 (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)+\int \frac {(c+d x) (a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)-\int (a+b \text {arctanh}(c+d x))d(c+d x)\right )+\int \frac {(c+d x) (a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (\int \frac {(c+d x) (a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)+b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (\int \frac {(c+d x) (a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2+b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (\int \frac {(a+b \text {arctanh}(c+d x))^2}{-c-d x+1}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2+b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (-2 b \int \frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))^2+b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (-2 b \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))^2+b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))^3-b \left (-2 b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {1}{2} (c+d x)^2 (a+b \text {arctanh}(c+d x))^2+\log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))^2+b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcTanh[c + d*x])^3)/3 - b*(-1/2*((c + d*x)^2*(a + b*ArcTanh[c + d*x])^2) - (a + b*ArcTanh[c + d*x])^3/(3*b) + (a + b*ArcT anh[c + d*x])^2*Log[2/(1 - c - d*x)] + b*(-(a*(c + d*x)) - b*(c + d*x)*Arc Tanh[c + d*x] + (a + b*ArcTanh[c + d*x])^2/(2*b) - (b*Log[1 - (c + d*x)^2] )/2) - 2*b*(-1/2*((a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)] ) + (b*PolyLog[3, 1 - 2/(1 - c - d*x)])/4))))/d
3.1.23.3.1 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_.)/((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[(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_) + (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]
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 = 3.06 (sec) , antiderivative size = 1079, normalized size of antiderivative = 4.10
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1079\) |
| default | \(\text {Expression too large to display}\) | \(1079\) |
| parts | \(\text {Expression too large to display}\) | \(1087\) |
1/d*(1/3*e^2*a^3*(d*x+c)^3+e^2*b^3*(1/3*(d*x+c)^3*arctanh(d*x+c)^3+1/2*(d* x+c)^2*arctanh(d*x+c)^2+1/2*arctanh(d*x+c)^2*ln(d*x+c-1)+1/2*arctanh(d*x+c )^2*ln(d*x+c+1)-arctanh(d*x+c)^2*ln((d*x+c+1)/(1-(d*x+c)^2)^(1/2))-arctanh (d*x+c)*polylog(2,-(d*x+c+1)^2/(1-(d*x+c)^2))+1/2*polylog(3,-(d*x+c+1)^2/( 1-(d*x+c)^2))-1/12*arctanh(d*x+c)*(6*I*arctanh(d*x+c)*Pi*csgn(I*(d*x+c+1)/ (1-(d*x+c)^2)^(1/2))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1))^2+6*I*arctanh(d*x+c )*Pi*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))^3+3*I*csgn(I*(d*x+c+1)^2/((d*x+ c)^2-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1)))^2*arctanh(d*x+c)*Pi*csgn(I/(1-(d*x+ c+1)^2/((d*x+c)^2-1)))+6*I*arctanh(d*x+c)*Pi-3*I*csgn(I*(d*x+c+1)^2/((d*x+ c)^2-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1)))^2*arctanh(d*x+c)*Pi*csgn(I*(d*x+c+1 )^2/((d*x+c)^2-1))+3*I*arctanh(d*x+c)*Pi*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)) ^3+3*I*arctanh(d*x+c)*Pi*csgn(I*(d*x+c+1)/(1-(d*x+c)^2)^(1/2))^2*csgn(I*(d *x+c+1)^2/((d*x+c)^2-1))-6*I*arctanh(d*x+c)*Pi*csgn(I/(1-(d*x+c+1)^2/((d*x +c)^2-1)))^2-3*I*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)/(1-(d*x+c+1)^2/((d*x+c)^ 2-1)))*arctanh(d*x+c)*Pi*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))*csgn(I*(d*x +c+1)^2/((d*x+c)^2-1))+3*I*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)/(1-(d*x+c+1)^2 /((d*x+c)^2-1)))^3*arctanh(d*x+c)*Pi-4*arctanh(d*x+c)^2+12*arctanh(d*x+c)* ln(2)+6*arctanh(d*x+c)-12*d*x-12*c-12)-ln(1+(d*x+c+1)^2/(1-(d*x+c)^2)))+3* e^2*a*b^2*(1/3*(d*x+c)^3*arctanh(d*x+c)^2+1/3*(d*x+c)^2*arctanh(d*x+c)+1/3 *arctanh(d*x+c)*ln(d*x+c-1)+1/3*arctanh(d*x+c)*ln(d*x+c+1)+1/3*d*x+1/3*...
\[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
integral(a^3*d^2*e^2*x^2 + 2*a^3*c*d*e^2*x + a^3*c^2*e^2 + (b^3*d^2*e^2*x^ 2 + 2*b^3*c*d*e^2*x + b^3*c^2*e^2)*arctanh(d*x + c)^3 + 3*(a*b^2*d^2*e^2*x ^2 + 2*a*b^2*c*d*e^2*x + a*b^2*c^2*e^2)*arctanh(d*x + c)^2 + 3*(a^2*b*d^2* e^2*x^2 + 2*a^2*b*c*d*e^2*x + a^2*b*c^2*e^2)*arctanh(d*x + c), x)
\[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx=e^{2} \left (\int a^{3} c^{2}\, dx + \int a^{3} d^{2} x^{2}\, dx + \int b^{3} c^{2} \operatorname {atanh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c^{2} \operatorname {atanh}{\left (c + d x \right )}\, dx + \int 2 a^{3} c d x\, dx + \int b^{3} d^{2} x^{2} \operatorname {atanh}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d^{2} x^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d^{2} x^{2} \operatorname {atanh}{\left (c + d x \right )}\, dx + \int 2 b^{3} c d x \operatorname {atanh}^{3}{\left (c + d x \right )}\, dx + \int 6 a b^{2} c d x \operatorname {atanh}^{2}{\left (c + d x \right )}\, dx + \int 6 a^{2} b c d x \operatorname {atanh}{\left (c + d x \right )}\, dx\right ) \]
e**2*(Integral(a**3*c**2, x) + Integral(a**3*d**2*x**2, x) + Integral(b**3 *c**2*atanh(c + d*x)**3, x) + Integral(3*a*b**2*c**2*atanh(c + d*x)**2, x) + Integral(3*a**2*b*c**2*atanh(c + d*x), x) + Integral(2*a**3*c*d*x, x) + Integral(b**3*d**2*x**2*atanh(c + d*x)**3, x) + Integral(3*a*b**2*d**2*x* *2*atanh(c + d*x)**2, x) + Integral(3*a**2*b*d**2*x**2*atanh(c + d*x), x) + Integral(2*b**3*c*d*x*atanh(c + d*x)**3, x) + Integral(6*a*b**2*c*d*x*at anh(c + d*x)**2, x) + Integral(6*a**2*b*c*d*x*atanh(c + d*x), x))
\[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/3*a^3*d^2*e^2*x^3 + a^3*c*d*e^2*x^2 + 3/2*(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*c*d*e^2 + 1/2*(2*x^3*arctanh(d*x + c) + d*((d*x^2 - 4*c *x)/d^3 + (c^3 + 3*c^2 + 3*c + 1)*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3* c - 1)*log(d*x + c - 1)/d^4))*a^2*b*d^2*e^2 + a^3*c^2*e^2*x + 3/2*(2*(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))*a^2*b*c^2*e^2/d - 1/24*((b^ 3*d^3*e^2*x^3 + 3*b^3*c*d^2*e^2*x^2 + 3*b^3*c^2*d*e^2*x + (c^3*e^2 - e^2)* b^3)*log(-d*x - c + 1)^3 - 3*(2*a*b^2*d^3*e^2*x^3 + (6*a*b^2*c*d^2*e^2 + b ^3*d^2*e^2)*x^2 + 2*(3*a*b^2*c^2*d*e^2 + b^3*c*d*e^2)*x + (b^3*d^3*e^2*x^3 + 3*b^3*c*d^2*e^2*x^2 + 3*b^3*c^2*d*e^2*x + (c^3*e^2 + e^2)*b^3)*log(d*x + c + 1))*log(-d*x - c + 1)^2)/d - integrate(-1/8*((b^3*d^3*e^2*x^3 + (3*c *d^2*e^2 - d^2*e^2)*b^3*x^2 + (3*c^2*d*e^2 - 2*c*d*e^2)*b^3*x + (c^3*e^2 - c^2*e^2)*b^3)*log(d*x + c + 1)^3 + 6*(a*b^2*d^3*e^2*x^3 + (3*c*d^2*e^2 - d^2*e^2)*a*b^2*x^2 + (3*c^2*d*e^2 - 2*c*d*e^2)*a*b^2*x + (c^3*e^2 - c^2*e^ 2)*a*b^2)*log(d*x + c + 1)^2 - (4*a*b^2*d^3*e^2*x^3 + 2*(6*a*b^2*c*d^2*e^2 + b^3*d^2*e^2)*x^2 + 3*(b^3*d^3*e^2*x^3 + (3*c*d^2*e^2 - d^2*e^2)*b^3*x^2 + (3*c^2*d*e^2 - 2*c*d*e^2)*b^3*x + (c^3*e^2 - c^2*e^2)*b^3)*log(d*x + c + 1)^2 + 4*(3*a*b^2*c^2*d*e^2 + b^3*c*d*e^2)*x + 2*(6*(c^3*e^2 - c^2*e^2)* a*b^2 + (c^3*e^2 + e^2)*b^3 + (6*a*b^2*d^3*e^2 + b^3*d^3*e^2)*x^3 + 3*(b^3 *c*d^2*e^2 + 2*(3*c*d^2*e^2 - d^2*e^2)*a*b^2)*x^2 + 3*(b^3*c^2*d*e^2 + ...
\[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^3 \,d x \]