Integrand size = 18, antiderivative size = 614 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\frac {3 a b^2 d e^2 x}{c^2}+\frac {b^3 e^3 x}{4 c^3}-\frac {b^3 e^3 \text {arctanh}(c x)}{4 c^4}+\frac {3 b^3 d e^2 x \text {arctanh}(c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \text {arctanh}(c x))}{4 c^2}-\frac {3 b d e^2 (a+b \text {arctanh}(c x))^2}{2 c^3}+\frac {b e^3 (a+b \text {arctanh}(c x))^2}{4 c^4}+\frac {3 b e \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{4 c^4}+\frac {3 b e \left (6 c^2 d^2+e^2\right ) x (a+b \text {arctanh}(c x))^2}{4 c^3}+\frac {3 b d e^2 x^2 (a+b \text {arctanh}(c x))^2}{2 c}+\frac {b e^3 x^3 (a+b \text {arctanh}(c x))^2}{4 c}+\frac {d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) (a+b \text {arctanh}(c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \text {arctanh}(c x))^3}{4 e}-\frac {b^2 e^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{2 c^4}-\frac {3 b d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {3 b^3 d e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {b^3 e^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 c^4}-\frac {3 b^3 e \left (6 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 c^4}-\frac {3 b^2 d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}+\frac {3 b^3 d \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3} \]
3*a*b^2*d*e^2*x/c^2+1/4*b^3*e^3*x/c^3-1/4*b^3*e^3*arctanh(c*x)/c^4+3*b^3*d *e^2*x*arctanh(c*x)/c^2+1/4*b^2*e^3*x^2*(a+b*arctanh(c*x))/c^2-3/2*b*d*e^2 *(a+b*arctanh(c*x))^2/c^3+1/4*b*e^3*(a+b*arctanh(c*x))^2/c^4+3/4*b*e*(6*c^ 2*d^2+e^2)*(a+b*arctanh(c*x))^2/c^4+3/4*b*e*(6*c^2*d^2+e^2)*x*(a+b*arctanh (c*x))^2/c^3+3/2*b*d*e^2*x^2*(a+b*arctanh(c*x))^2/c+1/4*b*e^3*x^3*(a+b*arc tanh(c*x))^2/c+d*(c^2*d^2+e^2)*(a+b*arctanh(c*x))^3/c^3-1/4*(c^4*d^4+6*c^2 *d^2*e^2+e^4)*(a+b*arctanh(c*x))^3/c^4/e+1/4*(e*x+d)^4*(a+b*arctanh(c*x))^ 3/e-1/2*b^2*e^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4-3/2*b^2*e*(6*c^2*d^2 +e^2)*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4-3*b*d*(c^2*d^2+e^2)*(a+b*arcta nh(c*x))^2*ln(2/(-c*x+1))/c^3+3/2*b^3*d*e^2*ln(-c^2*x^2+1)/c^3-1/4*b^3*e^3 *polylog(2,1-2/(-c*x+1))/c^4-3/4*b^3*e*(6*c^2*d^2+e^2)*polylog(2,1-2/(-c*x +1))/c^4-3*b^2*d*(c^2*d^2+e^2)*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/ c^3+3/2*b^3*d*(c^2*d^2+e^2)*polylog(3,1-2/(-c*x+1))/c^3
Time = 1.30 (sec) , antiderivative size = 830, normalized size of antiderivative = 1.35 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\frac {2 a^2 c \left (4 a c^3 d^3+3 b e \left (6 c^2 d^2+e^2\right )\right ) x+12 a^2 c^3 d e (a c d+b e) x^2+2 a^2 c^3 e^2 (4 a c d+b e) x^3+2 a^3 c^4 e^3 x^4+6 a^2 b c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \text {arctanh}(c x)+3 a^2 b \left (4 c^3 d^3+6 c^2 d^2 e+4 c d e^2+e^3\right ) \log (1-c x)+3 a^2 b \left (4 c^3 d^3-6 c^2 d^2 e+4 c d e^2-e^3\right ) \log (1+c x)+36 a b^2 c^2 d^2 e \left (2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+\log \left (1-c^2 x^2\right )\right )+2 a b^2 e^3 \left (-1+c^2 x^2+2 c x \left (3+c^2 x^2\right ) \text {arctanh}(c x)+3 \left (-1+c^4 x^4\right ) \text {arctanh}(c x)^2+4 \log \left (1-c^2 x^2\right )\right )-12 b^3 c^2 d^2 e \left (\text {arctanh}(c x) \left ((3-3 c x) \text {arctanh}(c x)+\left (1-c^2 x^2\right ) \text {arctanh}(c x)^2+6 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )-3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+24 a b^2 c d e^2 \left (c x+\left (-1+c^3 x^3\right ) \text {arctanh}(c x)^2+\text {arctanh}(c x) \left (-1+c^2 x^2-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+24 a b^2 c^3 d^3 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+2 b^3 e^3 \left (c x+\left (-4+3 c x+c^3 x^3\right ) \text {arctanh}(c x)^2+\left (-1+c^4 x^4\right ) \text {arctanh}(c x)^3+\text {arctanh}(c x) \left (-1+c^2 x^2-8 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+8 b^3 c^3 d^3 \left (\text {arctanh}(c x)^2 \left ((-1+c x) \text {arctanh}(c x)-3 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )+4 b^3 c d e^2 \left (6 c x \text {arctanh}(c x)-3 \text {arctanh}(c x)^2+3 c^2 x^2 \text {arctanh}(c x)^2-2 \text {arctanh}(c x)^3+2 c^3 x^3 \text {arctanh}(c x)^3-6 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+6 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )}{8 c^4} \]
(2*a^2*c*(4*a*c^3*d^3 + 3*b*e*(6*c^2*d^2 + e^2))*x + 12*a^2*c^3*d*e*(a*c*d + b*e)*x^2 + 2*a^2*c^3*e^2*(4*a*c*d + b*e)*x^3 + 2*a^3*c^4*e^3*x^4 + 6*a^ 2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x] + 3*a^2 *b*(4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d*e^2 + e^3)*Log[1 - c*x] + 3*a^2*b*(4*c ^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 - e^3)*Log[1 + c*x] + 36*a*b^2*c^2*d^2*e* (2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) + 2*a*b^2*e^3*(-1 + c^2*x^2 + 2*c*x*(3 + c^2*x^2)*ArcTanh[c*x] + 3*(-1 + c^4 *x^4)*ArcTanh[c*x]^2 + 4*Log[1 - c^2*x^2]) - 12*b^3*c^2*d^2*e*(ArcTanh[c*x ]*((3 - 3*c*x)*ArcTanh[c*x] + (1 - c^2*x^2)*ArcTanh[c*x]^2 + 6*Log[1 + E^( -2*ArcTanh[c*x])]) - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 24*a*b^2*c*d*e^ 2*(c*x + (-1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[c*x]*(-1 + c^2*x^2 - 2*Lo g[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 24*a*b^2 *c^3*d^3*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[ c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 2*b^3*e^3*(c*x + (-4 + 3*c*x + c^3*x^3)*ArcTanh[c*x]^2 + (-1 + c^4*x^4)*ArcTanh[c*x]^3 + ArcTanh[c*x]* (-1 + c^2*x^2 - 8*Log[1 + E^(-2*ArcTanh[c*x])]) + 4*PolyLog[2, -E^(-2*ArcT anh[c*x])]) + 8*b^3*c^3*d^3*(ArcTanh[c*x]^2*((-1 + c*x)*ArcTanh[c*x] - 3*L og[1 + E^(-2*ArcTanh[c*x])]) + 3*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c* x])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x])])/2) + 4*b^3*c*d*e^2*(6*c*x*ArcT anh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*ArcTanh[c*x]^2 - 2*ArcTanh[c*x]...
Time = 1.46 (sec) , antiderivative size = 608, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6480, 2009}
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.
| \(\displaystyle \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arctanh}(c x))^3}{4 e}-\frac {3 b c \int \left (-\frac {x^2 (a+b \text {arctanh}(c x))^2 e^4}{c^2}-\frac {4 d x (a+b \text {arctanh}(c x))^2 e^3}{c^2}-\frac {\left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2 e^2}{c^4}+\frac {\left (c^4 d^4+6 c^2 e^2 d^2+4 c^2 e \left (c^2 d^2+e^2\right ) x d+e^4\right ) (a+b \text {arctanh}(c x))^2}{c^4 \left (1-c^2 x^2\right )}\right )dx}{4 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arctanh}(c x))^3}{4 e}-\frac {3 b c \left (-\frac {e^4 (a+b \text {arctanh}(c x))^2}{3 c^5}+\frac {2 b e^4 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{3 c^5}+\frac {2 d e^3 (a+b \text {arctanh}(c x))^2}{c^4}-\frac {b e^4 x^2 (a+b \text {arctanh}(c x))}{3 c^3}-\frac {2 d e^3 x^2 (a+b \text {arctanh}(c x))^2}{c^2}-\frac {e^4 x^3 (a+b \text {arctanh}(c x))^2}{3 c^2}-\frac {e^2 \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{c^5}+\frac {2 b e^2 \left (6 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^5}+\frac {4 b d e \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^4}-\frac {e^2 x \left (6 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{c^4}-\frac {4 d e \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^3}{3 b c^4}+\frac {4 d e \left (c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c^4}+\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) (a+b \text {arctanh}(c x))^3}{3 b c^5}-\frac {4 a b d e^3 x}{c^3}+\frac {b^2 e^4 \text {arctanh}(c x)}{3 c^5}-\frac {4 b^2 d e^3 x \text {arctanh}(c x)}{c^3}+\frac {b^2 e^4 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^5}-\frac {b^2 e^4 x}{3 c^4}+\frac {b^2 e^2 \left (6 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^5}-\frac {2 b^2 d e \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c^4}-\frac {2 b^2 d e^3 \log \left (1-c^2 x^2\right )}{c^4}\right )}{4 e}\) |
((d + e*x)^4*(a + b*ArcTanh[c*x])^3)/(4*e) - (3*b*c*((-4*a*b*d*e^3*x)/c^3 - (b^2*e^4*x)/(3*c^4) + (b^2*e^4*ArcTanh[c*x])/(3*c^5) - (4*b^2*d*e^3*x*Ar cTanh[c*x])/c^3 - (b*e^4*x^2*(a + b*ArcTanh[c*x]))/(3*c^3) + (2*d*e^3*(a + b*ArcTanh[c*x])^2)/c^4 - (e^4*(a + b*ArcTanh[c*x])^2)/(3*c^5) - (e^2*(6*c ^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2)/c^5 - (e^2*(6*c^2*d^2 + e^2)*x*(a + b*ArcTanh[c*x])^2)/c^4 - (2*d*e^3*x^2*(a + b*ArcTanh[c*x])^2)/c^2 - (e^4*x ^3*(a + b*ArcTanh[c*x])^2)/(3*c^2) - (4*d*e*(c^2*d^2 + e^2)*(a + b*ArcTanh [c*x])^3)/(3*b*c^4) + ((c^4*d^4 + 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTanh[c*x] )^3)/(3*b*c^5) + (2*b*e^4*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^5) + (2*b*e^2*(6*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^5 + ( 4*d*e*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^4 - (2*b^ 2*d*e^3*Log[1 - c^2*x^2])/c^4 + (b^2*e^4*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c ^5) + (b^2*e^2*(6*c^2*d^2 + e^2)*PolyLog[2, 1 - 2/(1 - c*x)])/c^5 + (4*b*d *e*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^4 - (2*b^2*d*e*(c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/c^4))/(4*e)
3.1.15.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 ), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 37.83 (sec) , antiderivative size = 5093, normalized size of antiderivative = 8.29
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(5093\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(5124\) |
| default | \(\text {Expression too large to display}\) | \(5124\) |
\[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \]
integral(a^3*e^3*x^3 + 3*a^3*d*e^2*x^2 + 3*a^3*d^2*e*x + a^3*d^3 + (b^3*e^ 3*x^3 + 3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*arctanh(c*x)^3 + 3*(a*b ^2*e^3*x^3 + 3*a*b^2*d*e^2*x^2 + 3*a*b^2*d^2*e*x + a*b^2*d^3)*arctanh(c*x) ^2 + 3*(a^2*b*e^3*x^3 + 3*a^2*b*d*e^2*x^2 + 3*a^2*b*d^2*e*x + a^2*b*d^3)*a rctanh(c*x), x)
\[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{3}\, dx \]
\[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \]
1/4*a^3*e^3*x^4 + a^3*d*e^2*x^3 + 3/2*a^3*d^2*e*x^2 + 9/4*(2*x^2*arctanh(c *x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a^2*b*d^2*e + 3/2 *(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a^2*b*d*e^2 + 1 /8*(6*x^4*arctanh(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3 *log(c*x - 1)/c^5))*a^2*b*e^3 + a^3*d^3*x + 3/2*(2*c*x*arctanh(c*x) + log( -c^2*x^2 + 1))*a^2*b*d^3/c - 1/32*((b^3*c^4*e^3*x^4 + 4*b^3*c^4*d*e^2*x^3 + 6*b^3*c^4*d^2*e*x^2 + 4*b^3*c^4*d^3*x - (4*c^3*d^3 + 6*c^2*d^2*e + 4*c*d *e^2 + e^3)*b^3)*log(-c*x + 1)^3 - (6*a*b^2*c^4*e^3*x^4 + 2*(12*a*b^2*c^4* d*e^2 + b^3*c^3*e^3)*x^3 + 12*(3*a*b^2*c^4*d^2*e + b^3*c^3*d*e^2)*x^2 + 6* (4*a*b^2*c^4*d^3 + (6*c^3*d^2*e + c*e^3)*b^3)*x + 3*(b^3*c^4*e^3*x^4 + 4*b ^3*c^4*d*e^2*x^3 + 6*b^3*c^4*d^2*e*x^2 + 4*b^3*c^4*d^3*x + (4*c^3*d^3 - 6* c^2*d^2*e + 4*c*d*e^2 - e^3)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/c^4 - int egrate(-1/16*(2*(b^3*c^4*e^3*x^4 - b^3*c^3*d^3 + (3*c^4*d*e^2 - c^3*e^3)*b ^3*x^3 + 3*(c^4*d^2*e - c^3*d*e^2)*b^3*x^2 + (c^4*d^3 - 3*c^3*d^2*e)*b^3*x )*log(c*x + 1)^3 + 12*(a*b^2*c^4*e^3*x^4 - a*b^2*c^3*d^3 + (3*c^4*d*e^2 - c^3*e^3)*a*b^2*x^3 + 3*(c^4*d^2*e - c^3*d*e^2)*a*b^2*x^2 + (c^4*d^3 - 3*c^ 3*d^2*e)*a*b^2*x)*log(c*x + 1)^2 - (6*a*b^2*c^4*e^3*x^4 + 2*(12*a*b^2*c^4* d*e^2 + b^3*c^3*e^3)*x^3 + 12*(3*a*b^2*c^4*d^2*e + b^3*c^3*d*e^2)*x^2 + 6* (b^3*c^4*e^3*x^4 - b^3*c^3*d^3 + (3*c^4*d*e^2 - c^3*e^3)*b^3*x^3 + 3*(c^4* d^2*e - c^3*d*e^2)*b^3*x^2 + (c^4*d^3 - 3*c^3*d^2*e)*b^3*x)*log(c*x + 1...
\[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^3 \,d x \]