Integrand size = 23, antiderivative size = 159 \[ \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {1}{2} a b e^3 x+\frac {b^2 e^3 (c+d x)^2}{12 d}+\frac {b^2 e^3 (c+d x) \text {arctanh}(c+d x)}{2 d}+\frac {b e^3 (c+d x)^3 (a+b \text {arctanh}(c+d x))}{6 d}-\frac {e^3 (a+b \text {arctanh}(c+d x))^2}{4 d}+\frac {e^3 (c+d x)^4 (a+b \text {arctanh}(c+d x))^2}{4 d}+\frac {b^2 e^3 \log \left (1-(c+d x)^2\right )}{3 d} \]
1/2*a*b*e^3*x+1/12*b^2*e^3*(d*x+c)^2/d+1/2*b^2*e^3*(d*x+c)*arctanh(d*x+c)/ d+1/6*b*e^3*(d*x+c)^3*(a+b*arctanh(d*x+c))/d-1/4*e^3*(a+b*arctanh(d*x+c))^ 2/d+1/4*e^3*(d*x+c)^4*(a+b*arctanh(d*x+c))^2/d+1/3*b^2*e^3*ln(1-(d*x+c)^2) /d
Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.93 \[ \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {e^3 \left (6 a b (c+d x)+b^2 (c+d x)^2+2 a b (c+d x)^3+3 a^2 (c+d x)^4+2 b (c+d x) \left (3 b+b (c+d x)^2+3 a (c+d x)^3\right ) \text {arctanh}(c+d x)+3 b^2 \left (-1+(c+d x)^4\right ) \text {arctanh}(c+d x)^2+b (3 a+4 b) \log (1-c-d x)+b (-3 a+4 b) \log (1+c+d x)\right )}{12 d} \]
(e^3*(6*a*b*(c + d*x) + b^2*(c + d*x)^2 + 2*a*b*(c + d*x)^3 + 3*a^2*(c + d *x)^4 + 2*b*(c + d*x)*(3*b + b*(c + d*x)^2 + 3*a*(c + d*x)^3)*ArcTanh[c + d*x] + 3*b^2*(-1 + (c + d*x)^4)*ArcTanh[c + d*x]^2 + b*(3*a + 4*b)*Log[1 - c - d*x] + b*(-3*a + 4*b)*Log[1 + c + d*x]))/(12*d)
Time = 0.85 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.89, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6657, 27, 6452, 6542, 6452, 243, 49, 2009, 6542, 2009, 6510}
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.
| \(\displaystyle \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 6657 |
| \(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \text {arctanh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \text {arctanh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \int \frac {(c+d x)^4 (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (\int \frac {(c+d x)^2 (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)-\int (c+d x)^2 (a+b \text {arctanh}(c+d x))d(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (\int \frac {(c+d x)^2 (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)+\frac {1}{3} b \int \frac {(c+d x)^3}{1-(c+d x)^2}d(c+d x)-\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (\int \frac {(c+d x)^2 (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)+\frac {1}{6} b \int \frac {(c+d x)^2}{-c-d x+1}d(c+d x)^2-\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (\int \frac {(c+d x)^2 (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)+\frac {1}{6} b \int \left (\frac {1}{-c-d x+1}-1\right )d(c+d x)^2-\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (\int \frac {(c+d x)^2 (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}d(c+d x)-\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))+\frac {1}{6} b (-\log (-c-d x+1)-c-d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (-\int (a+b \text {arctanh}(c+d x))d(c+d x)+\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)-\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))+\frac {1}{6} b (-\log (-c-d x+1)-c-d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)-\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)+\frac {1}{6} b (-\log (-c-d x+1)-c-d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arctanh}(c+d x))^2-\frac {1}{2} b \left (-\frac {1}{3} (c+d x)^3 (a+b \text {arctanh}(c+d x))+\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-a (c+d x)-b (c+d x) \text {arctanh}(c+d x)+\frac {1}{6} b (-\log (-c-d x+1)-c-d x)-\frac {1}{2} b \log \left (1-(c+d x)^2\right )\right )\right )}{d}\) |
(e^3*(((c + d*x)^4*(a + b*ArcTanh[c + d*x])^2)/4 - (b*(-(a*(c + d*x)) - b* (c + d*x)*ArcTanh[c + d*x] - ((c + d*x)^3*(a + b*ArcTanh[c + d*x]))/3 + (a + b*ArcTanh[c + d*x])^2/(2*b) + (b*(-c - d*x - Log[1 - c - d*x]))/6 - (b* Log[1 - (c + d*x)^2])/2))/2))/d
3.1.15.3.1 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_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_)^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_) + (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]
Time = 0.36 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.64
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arctanh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )}{6}+\frac {\left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (d x +c -1\right )^{2}}{16}+\frac {\ln \left (d x +c +1\right )^{2}}{16}-\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}+\frac {\left (d x +c \right )^{2}}{12}+\frac {\ln \left (d x +c -1\right )}{3}+\frac {\ln \left (d x +c +1\right )}{3}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arctanh}\left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3}}{12}+\frac {d x}{4}+\frac {c}{4}+\frac {\ln \left (d x +c -1\right )}{8}-\frac {\ln \left (d x +c +1\right )}{8}\right )}{d}\) | \(261\) |
| default | \(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arctanh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )}{6}+\frac {\left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (d x +c -1\right )^{2}}{16}+\frac {\ln \left (d x +c +1\right )^{2}}{16}-\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}+\frac {\left (d x +c \right )^{2}}{12}+\frac {\ln \left (d x +c -1\right )}{3}+\frac {\ln \left (d x +c +1\right )}{3}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arctanh}\left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3}}{12}+\frac {d x}{4}+\frac {c}{4}+\frac {\ln \left (d x +c -1\right )}{8}-\frac {\ln \left (d x +c +1\right )}{8}\right )}{d}\) | \(261\) |
| parts | \(\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arctanh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )}{6}+\frac {\left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (d x +c -1\right )^{2}}{16}+\frac {\ln \left (d x +c +1\right )^{2}}{16}-\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}+\frac {\left (d x +c \right )^{2}}{12}+\frac {\ln \left (d x +c -1\right )}{3}+\frac {\ln \left (d x +c +1\right )}{3}\right )}{d}+\frac {2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arctanh}\left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3}}{12}+\frac {d x}{4}+\frac {c}{4}+\frac {\ln \left (d x +c -1\right )}{8}-\frac {\ln \left (d x +c +1\right )}{8}\right )}{d}\) | \(266\) |
| parallelrisch | \(\frac {18 d \,e^{3} c^{2} a^{2}-5 d \,e^{3} c^{2} b^{2}+e^{3} b^{2} d -18 a b \,c^{3} d \,e^{3}-6 a b c d \,e^{3}+12 x^{3} a^{2} c \,d^{4} e^{3}+2 x^{3} a b \,d^{4} e^{3}+6 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} d^{2} e^{3}+12 x \,a^{2} c^{3} d^{2} e^{3}+6 x a b \,d^{2} e^{3}+2 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{3} d \,e^{3}+6 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c d \,e^{3}-6 \,\operatorname {arctanh}\left (d x +c \right ) a b d \,e^{3}+2 x^{3} \operatorname {arctanh}\left (d x +c \right ) b^{2} d^{4} e^{3}+3 \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{4} d \,e^{3}+18 x^{2} a^{2} c^{2} d^{3} e^{3}+3 d^{5} e^{3} b^{2} \operatorname {arctanh}\left (d x +c \right )^{2} x^{4}+2 x \,b^{2} c \,d^{2} e^{3}+3 x^{4} a^{2} d^{5} e^{3}+8 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} d \,e^{3}-3 e^{3} b^{2} \operatorname {arctanh}\left (d x +c \right )^{2} d +x^{2} b^{2} d^{3} e^{3}+8 \ln \left (d x +c -1\right ) b^{2} d \,e^{3}+24 x \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{3} d^{2} e^{3}+36 x^{2} \operatorname {arctanh}\left (d x +c \right ) a b \,c^{2} d^{3} e^{3}+24 x^{3} \operatorname {arctanh}\left (d x +c \right ) a b c \,d^{4} e^{3}-42 a^{2} c^{4} d \,e^{3}+12 d^{4} e^{3} c \,b^{2} x^{3} \operatorname {arctanh}\left (d x +c \right )^{2}+6 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{2} d^{2} e^{3}+6 x^{2} \operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{3} e^{3}+6 x a b \,c^{2} d^{2} e^{3}+6 \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{4} d \,e^{3}+12 x \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{3} d^{2} e^{3}+18 x^{2} \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{2} d^{3} e^{3}+6 x^{4} \operatorname {arctanh}\left (d x +c \right ) a b \,d^{5} e^{3}+6 x^{2} a b c \,d^{3} e^{3}}{12 d^{2}}\) | \(597\) |
| risch | \(-e^{3} d^{2} a b c \,x^{3} \ln \left (-d x -c +1\right )-\frac {3 e^{3} d a b \,c^{2} x^{2} \ln \left (-d x -c +1\right )}{2}+\frac {a b \,e^{3} x}{2}+e^{3} d^{2} a^{2} c \,x^{3}+\frac {3 e^{3} d \,a^{2} c^{2} x^{2}}{2}+\frac {e^{3} d^{2} a b \,x^{3}}{6}+e^{3} a^{2} c^{3} x +\frac {e^{3} b^{2} c x}{6}+\frac {e^{3} d^{3} b^{2} x^{4} \ln \left (-d x -c +1\right )^{2}}{16}-\frac {e^{3} d^{2} b^{2} x^{3} \ln \left (-d x -c +1\right )}{12}+\frac {e^{3} b^{2} c^{4} \ln \left (-d x -c +1\right )^{2}}{16 d}-\frac {e^{3} \ln \left (d x +c -1\right ) b^{2} c^{3}}{12 d}+\frac {e^{3} \ln \left (-d x -c -1\right ) b^{2} c^{3}}{12 d}-\frac {e^{3} \ln \left (d x +c -1\right ) b^{2} c}{4 d}+\frac {e^{3} \ln \left (-d x -c -1\right ) b^{2} c}{4 d}+\frac {e^{3} \ln \left (d x +c -1\right ) a b}{4 d}-\frac {e^{3} \ln \left (-d x -c -1\right ) a b}{4 d}+\frac {e^{3} b^{2} c^{3} x \ln \left (-d x -c +1\right )^{2}}{4}-\frac {e^{3} b^{2} c^{2} x \ln \left (-d x -c +1\right )}{4}+\frac {e^{3} b^{2} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}-1\right ) \ln \left (d x +c +1\right )^{2}}{16 d}+\frac {e^{3} b \left (-3 b \,d^{4} x^{4} \ln \left (-d x -c +1\right )+6 a \,d^{4} x^{4}-12 b c \,d^{3} x^{3} \ln \left (-d x -c +1\right )+24 a c \,d^{3} x^{3}-18 b \,c^{2} d^{2} x^{2} \ln \left (-d x -c +1\right )+36 a \,c^{2} d^{2} x^{2}-12 b \,c^{3} d x \ln \left (-d x -c +1\right )+2 b \,d^{3} x^{3}+24 a \,c^{3} d x -3 b \,c^{4} \ln \left (-d x -c +1\right )+6 b c \,d^{2} x^{2}+6 b \,c^{2} d x +6 b d x +3 b \ln \left (-d x -c +1\right )\right ) \ln \left (d x +c +1\right )}{24 d}+\frac {e^{3} d^{3} a^{2} x^{4}}{4}+\frac {e^{3} d \,b^{2} x^{2}}{12}-\frac {e^{3} b^{2} x \ln \left (-d x -c +1\right )}{4}-\frac {e^{3} b^{2} \ln \left (-d x -c +1\right )^{2}}{16 d}+\frac {e^{3} \ln \left (-d x -c -1\right ) b^{2}}{3 d}+\frac {e^{3} b^{2} \ln \left (d x +c -1\right )}{3 d}+\frac {e^{3} d a b c \,x^{2}}{2}+\frac {e^{3} a b \,c^{2} x}{2}-e^{3} a b \,c^{3} x \ln \left (-d x -c +1\right )+\frac {e^{3} d^{2} b^{2} c \,x^{3} \ln \left (-d x -c +1\right )^{2}}{4}-\frac {e^{3} d^{3} a b \,x^{4} \ln \left (-d x -c +1\right )}{4}+\frac {3 e^{3} d \,b^{2} c^{2} x^{2} \ln \left (-d x -c +1\right )^{2}}{8}-\frac {e^{3} d \,b^{2} c \,x^{2} \ln \left (-d x -c +1\right )}{4}-\frac {e^{3} \ln \left (d x +c -1\right ) a b \,c^{4}}{4 d}+\frac {e^{3} \ln \left (-d x -c -1\right ) a b \,c^{4}}{4 d}\) | \(915\) |
1/d*(1/4*e^3*a^2*(d*x+c)^4+e^3*b^2*(1/4*(d*x+c)^4*arctanh(d*x+c)^2+1/6*(d* x+c)^3*arctanh(d*x+c)+1/2*(d*x+c)*arctanh(d*x+c)+1/4*arctanh(d*x+c)*ln(d*x +c-1)-1/4*arctanh(d*x+c)*ln(d*x+c+1)-1/8*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2) +1/16*ln(d*x+c-1)^2+1/16*ln(d*x+c+1)^2-1/8*(ln(d*x+c+1)-ln(1/2*d*x+1/2*c+1 /2))*ln(-1/2*d*x-1/2*c+1/2)+1/12*(d*x+c)^2+1/3*ln(d*x+c-1)+1/3*ln(d*x+c+1) )+2*e^3*a*b*(1/4*(d*x+c)^4*arctanh(d*x+c)+1/12*(d*x+c)^3+1/4*d*x+1/4*c+1/8 *ln(d*x+c-1)-1/8*ln(d*x+c+1)))
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (145) = 290\).
Time = 0.27 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.41 \[ \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {12 \, a^{2} d^{4} e^{3} x^{4} + 8 \, {\left (6 \, a^{2} c + a b\right )} d^{3} e^{3} x^{3} + 4 \, {\left (18 \, a^{2} c^{2} + 6 \, a b c + b^{2}\right )} d^{2} e^{3} x^{2} + 8 \, {\left (6 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c + 3 \, a b\right )} d e^{3} x + 4 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b + 4 \, b^{2}\right )} e^{3} \log \left (d x + c + 1\right ) - 4 \, {\left (3 \, a b c^{4} + b^{2} c^{3} + 3 \, b^{2} c - 3 \, a b - 4 \, b^{2}\right )} e^{3} \log \left (d x + c - 1\right ) + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (b^{2} c^{4} - b^{2}\right )} e^{3}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2} + 4 \, {\left (3 \, a b d^{4} e^{3} x^{4} + {\left (12 \, a b c + b^{2}\right )} d^{3} e^{3} x^{3} + 3 \, {\left (6 \, a b c^{2} + b^{2} c\right )} d^{2} e^{3} x^{2} + 3 \, {\left (4 \, a b c^{3} + b^{2} c^{2} + b^{2}\right )} d e^{3} x\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{48 \, d} \]
1/48*(12*a^2*d^4*e^3*x^4 + 8*(6*a^2*c + a*b)*d^3*e^3*x^3 + 4*(18*a^2*c^2 + 6*a*b*c + b^2)*d^2*e^3*x^2 + 8*(6*a^2*c^3 + 3*a*b*c^2 + b^2*c + 3*a*b)*d* e^3*x + 4*(3*a*b*c^4 + b^2*c^3 + 3*b^2*c - 3*a*b + 4*b^2)*e^3*log(d*x + c + 1) - 4*(3*a*b*c^4 + b^2*c^3 + 3*b^2*c - 3*a*b - 4*b^2)*e^3*log(d*x + c - 1) + 3*(b^2*d^4*e^3*x^4 + 4*b^2*c*d^3*e^3*x^3 + 6*b^2*c^2*d^2*e^3*x^2 + 4 *b^2*c^3*d*e^3*x + (b^2*c^4 - b^2)*e^3)*log(-(d*x + c + 1)/(d*x + c - 1))^ 2 + 4*(3*a*b*d^4*e^3*x^4 + (12*a*b*c + b^2)*d^3*e^3*x^3 + 3*(6*a*b*c^2 + b ^2*c)*d^2*e^3*x^2 + 3*(4*a*b*c^3 + b^2*c^2 + b^2)*d*e^3*x)*log(-(d*x + c + 1)/(d*x + c - 1)))/d
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (138) = 276\).
Time = 0.68 (sec) , antiderivative size = 581, normalized size of antiderivative = 3.65 \[ \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\begin {cases} a^{2} c^{3} e^{3} x + \frac {3 a^{2} c^{2} d e^{3} x^{2}}{2} + a^{2} c d^{2} e^{3} x^{3} + \frac {a^{2} d^{3} e^{3} x^{4}}{4} + \frac {a b c^{4} e^{3} \operatorname {atanh}{\left (c + d x \right )}}{2 d} + 2 a b c^{3} e^{3} x \operatorname {atanh}{\left (c + d x \right )} + 3 a b c^{2} d e^{3} x^{2} \operatorname {atanh}{\left (c + d x \right )} + \frac {a b c^{2} e^{3} x}{2} + 2 a b c d^{2} e^{3} x^{3} \operatorname {atanh}{\left (c + d x \right )} + \frac {a b c d e^{3} x^{2}}{2} + \frac {a b d^{3} e^{3} x^{4} \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {a b d^{2} e^{3} x^{3}}{6} + \frac {a b e^{3} x}{2} - \frac {a b e^{3} \operatorname {atanh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{4} e^{3} \operatorname {atanh}^{2}{\left (c + d x \right )}}{4 d} + b^{2} c^{3} e^{3} x \operatorname {atanh}^{2}{\left (c + d x \right )} + \frac {b^{2} c^{3} e^{3} \operatorname {atanh}{\left (c + d x \right )}}{6 d} + \frac {3 b^{2} c^{2} d e^{3} x^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{2} + \frac {b^{2} c^{2} e^{3} x \operatorname {atanh}{\left (c + d x \right )}}{2} + b^{2} c d^{2} e^{3} x^{3} \operatorname {atanh}^{2}{\left (c + d x \right )} + \frac {b^{2} c d e^{3} x^{2} \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {b^{2} c e^{3} x}{6} + \frac {b^{2} c e^{3} \operatorname {atanh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d^{3} e^{3} x^{4} \operatorname {atanh}^{2}{\left (c + d x \right )}}{4} + \frac {b^{2} d^{2} e^{3} x^{3} \operatorname {atanh}{\left (c + d x \right )}}{6} + \frac {b^{2} d e^{3} x^{2}}{12} + \frac {b^{2} e^{3} x \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {2 b^{2} e^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{3 d} - \frac {b^{2} e^{3} \operatorname {atanh}^{2}{\left (c + d x \right )}}{4 d} - \frac {2 b^{2} e^{3} \operatorname {atanh}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname {atanh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**3*e**3*x + 3*a**2*c**2*d*e**3*x**2/2 + a**2*c*d**2*e**3 *x**3 + a**2*d**3*e**3*x**4/4 + a*b*c**4*e**3*atanh(c + d*x)/(2*d) + 2*a*b *c**3*e**3*x*atanh(c + d*x) + 3*a*b*c**2*d*e**3*x**2*atanh(c + d*x) + a*b* c**2*e**3*x/2 + 2*a*b*c*d**2*e**3*x**3*atanh(c + d*x) + a*b*c*d*e**3*x**2/ 2 + a*b*d**3*e**3*x**4*atanh(c + d*x)/2 + a*b*d**2*e**3*x**3/6 + a*b*e**3* x/2 - a*b*e**3*atanh(c + d*x)/(2*d) + b**2*c**4*e**3*atanh(c + d*x)**2/(4* d) + b**2*c**3*e**3*x*atanh(c + d*x)**2 + b**2*c**3*e**3*atanh(c + d*x)/(6 *d) + 3*b**2*c**2*d*e**3*x**2*atanh(c + d*x)**2/2 + b**2*c**2*e**3*x*atanh (c + d*x)/2 + b**2*c*d**2*e**3*x**3*atanh(c + d*x)**2 + b**2*c*d*e**3*x**2 *atanh(c + d*x)/2 + b**2*c*e**3*x/6 + b**2*c*e**3*atanh(c + d*x)/(2*d) + b **2*d**3*e**3*x**4*atanh(c + d*x)**2/4 + b**2*d**2*e**3*x**3*atanh(c + d*x )/6 + b**2*d*e**3*x**2/12 + b**2*e**3*x*atanh(c + d*x)/2 + 2*b**2*e**3*log (c/d + x + 1/d)/(3*d) - b**2*e**3*atanh(c + d*x)**2/(4*d) - 2*b**2*e**3*at anh(c + d*x)/(3*d), Ne(d, 0)), (c**3*e**3*x*(a + b*atanh(c))**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (145) = 290\).
Time = 0.39 (sec) , antiderivative size = 827, normalized size of antiderivative = 5.20 \[ \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {1}{4} \, a^{2} d^{3} e^{3} x^{4} + a^{2} c d^{2} e^{3} x^{3} + \frac {3}{2} \, a^{2} c^{2} d e^{3} x^{2} + \frac {3}{2} \, {\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 b c^{2} d e^{3} + {\left (2 \, x^{3} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} a b c d^{2} e^{3} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, {\left (d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} + 1\right )} x\right )}}{d^{4}} - \frac {3 \, {\left (c^{4} + 4 \, c^{3} + 6 \, c^{2} + 4 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{5}} + \frac {3 \, {\left (c^{4} - 4 \, c^{3} + 6 \, c^{2} - 4 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{5}}\right )}\right )} a b d^{3} e^{3} + a^{2} c^{3} e^{3} x + \frac {{\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 b c^{3} e^{3}}{d} + \frac {4 \, b^{2} d^{2} e^{3} x^{2} + 8 \, b^{2} c d e^{3} x + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (c^{4} e^{3} - e^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (c^{4} e^{3} - e^{3}\right )} b^{2}\right )} \log \left (-d x - c + 1\right )^{2} + 4 \, {\left (b^{2} d^{3} e^{3} x^{3} + 3 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (c^{2} d e^{3} + d e^{3}\right )} b^{2} x + {\left (c^{3} e^{3} + 3 \, c e^{3} + 4 \, e^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (2 \, b^{2} d^{3} e^{3} x^{3} + 6 \, b^{2} c d^{2} e^{3} x^{2} + 6 \, {\left (c^{2} d e^{3} + d e^{3}\right )} b^{2} x + 2 \, {\left (c^{3} e^{3} + 3 \, c e^{3} - 4 \, e^{3}\right )} b^{2} + 3 \, {\left (b^{2} d^{4} e^{3} x^{4} + 4 \, b^{2} c d^{3} e^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 4 \, b^{2} c^{3} d e^{3} x + {\left (c^{4} e^{3} - e^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{48 \, d} \]
1/4*a^2*d^3*e^3*x^4 + a^2*c*d^2*e^3*x^3 + 3/2*a^2*c^2*d*e^3*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*b*c^2*d*e^3 + (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*b*c*d^2*e^3 + 1/12*( 6*x^4*arctanh(d*x + c) + d*(2*(d^2*x^3 - 3*c*d*x^2 + 3*(3*c^2 + 1)*x)/d^4 - 3*(c^4 + 4*c^3 + 6*c^2 + 4*c + 1)*log(d*x + c + 1)/d^5 + 3*(c^4 - 4*c^3 + 6*c^2 - 4*c + 1)*log(d*x + c - 1)/d^5))*a*b*d^3*e^3 + a^2*c^3*e^3*x + (2 *(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*c^3*e^3/d + 1/48* (4*b^2*d^2*e^3*x^2 + 8*b^2*c*d*e^3*x + 3*(b^2*d^4*e^3*x^4 + 4*b^2*c*d^3*e^ 3*x^3 + 6*b^2*c^2*d^2*e^3*x^2 + 4*b^2*c^3*d*e^3*x + (c^4*e^3 - e^3)*b^2)*l og(d*x + c + 1)^2 + 3*(b^2*d^4*e^3*x^4 + 4*b^2*c*d^3*e^3*x^3 + 6*b^2*c^2*d ^2*e^3*x^2 + 4*b^2*c^3*d*e^3*x + (c^4*e^3 - e^3)*b^2)*log(-d*x - c + 1)^2 + 4*(b^2*d^3*e^3*x^3 + 3*b^2*c*d^2*e^3*x^2 + 3*(c^2*d*e^3 + d*e^3)*b^2*x + (c^3*e^3 + 3*c*e^3 + 4*e^3)*b^2)*log(d*x + c + 1) - 2*(2*b^2*d^3*e^3*x^3 + 6*b^2*c*d^2*e^3*x^2 + 6*(c^2*d*e^3 + d*e^3)*b^2*x + 2*(c^3*e^3 + 3*c*e^3 - 4*e^3)*b^2 + 3*(b^2*d^4*e^3*x^4 + 4*b^2*c*d^3*e^3*x^3 + 6*b^2*c^2*d^2*e ^3*x^2 + 4*b^2*c^3*d*e^3*x + (c^4*e^3 - e^3)*b^2)*log(d*x + c + 1))*log(-d *x - c + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (145) = 290\).
Time = 0.32 (sec) , antiderivative size = 733, normalized size of antiderivative = 4.61 \[ \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=-\frac {1}{12} \, {\left (\frac {4 \, b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right )}{d^{2}} - \frac {4 \, b^{2} e^{3} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d^{2}} - \frac {3 \, {\left (\frac {{\left (d x + c + 1\right )}^{3} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} + \frac {{\left (d x + c + 1\right )} b^{2} e^{3}}{d x + c - 1}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{\frac {{\left (d x + c + 1\right )}^{4} d^{2}}{{\left (d x + c - 1\right )}^{4}} - \frac {4 \, {\left (d x + c + 1\right )}^{3} d^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} d^{2}}{{\left (d x + c - 1\right )}^{2}} - \frac {4 \, {\left (d x + c + 1\right )} d^{2}}{d x + c - 1} + d^{2}} - \frac {2 \, {\left (\frac {6 \, {\left (d x + c + 1\right )}^{3} a b e^{3}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )} a b e^{3}}{d x + c - 1} + \frac {3 \, {\left (d x + c + 1\right )}^{3} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {6 \, {\left (d x + c + 1\right )}^{2} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {5 \, {\left (d x + c + 1\right )} b^{2} e^{3}}{d x + c - 1} - 2 \, b^{2} e^{3}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{\frac {{\left (d x + c + 1\right )}^{4} d^{2}}{{\left (d x + c - 1\right )}^{4}} - \frac {4 \, {\left (d x + c + 1\right )}^{3} d^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} d^{2}}{{\left (d x + c - 1\right )}^{2}} - \frac {4 \, {\left (d x + c + 1\right )} d^{2}}{d x + c - 1} + d^{2}} - \frac {2 \, {\left (\frac {6 \, {\left (d x + c + 1\right )}^{3} a^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )} a^{2} e^{3}}{d x + c - 1} + \frac {6 \, {\left (d x + c + 1\right )}^{3} a b e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {12 \, {\left (d x + c + 1\right )}^{2} a b e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {10 \, {\left (d x + c + 1\right )} a b e^{3}}{d x + c - 1} - 4 \, a b e^{3} + \frac {{\left (d x + c + 1\right )}^{3} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{3}} - \frac {2 \, {\left (d x + c + 1\right )}^{2} b^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {{\left (d x + c + 1\right )} b^{2} e^{3}}{d x + c - 1}\right )}}{\frac {{\left (d x + c + 1\right )}^{4} d^{2}}{{\left (d x + c - 1\right )}^{4}} - \frac {4 \, {\left (d x + c + 1\right )}^{3} d^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} d^{2}}{{\left (d x + c - 1\right )}^{2}} - \frac {4 \, {\left (d x + c + 1\right )} d^{2}}{d x + c - 1} + d^{2}}\right )} {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} \]
-1/12*(4*b^2*e^3*log(-(d*x + c + 1)/(d*x + c - 1) + 1)/d^2 - 4*b^2*e^3*log (-(d*x + c + 1)/(d*x + c - 1))/d^2 - 3*((d*x + c + 1)^3*b^2*e^3/(d*x + c - 1)^3 + (d*x + c + 1)*b^2*e^3/(d*x + c - 1))*log(-(d*x + c + 1)/(d*x + c - 1))^2/((d*x + c + 1)^4*d^2/(d*x + c - 1)^4 - 4*(d*x + c + 1)^3*d^2/(d*x + c - 1)^3 + 6*(d*x + c + 1)^2*d^2/(d*x + c - 1)^2 - 4*(d*x + c + 1)*d^2/(d *x + c - 1) + d^2) - 2*(6*(d*x + c + 1)^3*a*b*e^3/(d*x + c - 1)^3 + 6*(d*x + c + 1)*a*b*e^3/(d*x + c - 1) + 3*(d*x + c + 1)^3*b^2*e^3/(d*x + c - 1)^ 3 - 6*(d*x + c + 1)^2*b^2*e^3/(d*x + c - 1)^2 + 5*(d*x + c + 1)*b^2*e^3/(d *x + c - 1) - 2*b^2*e^3)*log(-(d*x + c + 1)/(d*x + c - 1))/((d*x + c + 1)^ 4*d^2/(d*x + c - 1)^4 - 4*(d*x + c + 1)^3*d^2/(d*x + c - 1)^3 + 6*(d*x + c + 1)^2*d^2/(d*x + c - 1)^2 - 4*(d*x + c + 1)*d^2/(d*x + c - 1) + d^2) - 2 *(6*(d*x + c + 1)^3*a^2*e^3/(d*x + c - 1)^3 + 6*(d*x + c + 1)*a^2*e^3/(d*x + c - 1) + 6*(d*x + c + 1)^3*a*b*e^3/(d*x + c - 1)^3 - 12*(d*x + c + 1)^2 *a*b*e^3/(d*x + c - 1)^2 + 10*(d*x + c + 1)*a*b*e^3/(d*x + c - 1) - 4*a*b* e^3 + (d*x + c + 1)^3*b^2*e^3/(d*x + c - 1)^3 - 2*(d*x + c + 1)^2*b^2*e^3/ (d*x + c - 1)^2 + (d*x + c + 1)*b^2*e^3/(d*x + c - 1))/((d*x + c + 1)^4*d^ 2/(d*x + c - 1)^4 - 4*(d*x + c + 1)^3*d^2/(d*x + c - 1)^3 + 6*(d*x + c + 1 )^2*d^2/(d*x + c - 1)^2 - 4*(d*x + c + 1)*d^2/(d*x + c - 1) + d^2))*((c + 1)*d - (c - 1)*d)
Time = 5.39 (sec) , antiderivative size = 1730, normalized size of antiderivative = 10.88 \[ \int (c e+d e x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\text {Too large to display} \]
log(1 - d*x - c)^2*((b^2*c^3*e^3*x)/4 - (b^2*e^3 - b^2*c^4*e^3)/(16*d) + ( b^2*d^3*e^3*x^4)/16 + (3*b^2*c^2*d*e^3*x^2)/8 + (b^2*c*d^2*e^3*x^3)/4) + x *((c*e^3*(b^2 - 6*a^2 + 20*a^2*c^2 + 6*a*b*c))/2 + ((6*c^2 - 6)*(2*a^2*c*d ^2*e^3 - (a*d^2*e^3*(b + 10*a*c))/2))/(6*d^2) - (2*c*((2*c*(2*a^2*c*d^2*e^ 3 - (a*d^2*e^3*(b + 10*a*c))/2))/d + (d*e^3*(b^2 - 6*a^2 + 60*a^2*c^2 + 12 *a*b*c))/6 - (a^2*d*e^3*(6*c^2 - 6))/6))/d) - log(1 - d*x - c)*(log(c + d* x + 1)*((b^2*c^3*e^3*x)/2 - ((b^2*e^3)/2 - (b^2*c^4*e^3)/2)/(4*d) + (b^2*d ^3*e^3*x^4)/8 + (3*b^2*c^2*d*e^3*x^2)/4 + (b^2*c*d^2*e^3*x^3)/2) + (x^2*(( (d*(c - 1) + d*(c + 1))*(32*b^2*c*d^4*e^3 - 8*b^2*d^3*e^3*(d*(c - 1) + d*( c + 1)) + 8*b^2*d^4*e^3*(c - 1)))/d^2 - 48*b^2*c^2*d^3*e^3 + 8*b^2*d^3*e^3 *(c - 1)*(c + 1) - 32*b^2*c*d^3*e^3*(c - 1)))/(128*d^2) - (x^2*(((d*(c - 1 ) + d*(c + 1))*(32*b*d^4*e^3*(8*a*c - 2*a + b*c) - 8*b*d^3*e^3*(d*(c - 1) + d*(c + 1))*(8*a + b) + 8*b*d^4*e^3*(8*a + b)*(c + 1)))/d^2 - 48*b*c*d^3* e^3*(8*a*c - 4*a + b*c) - 32*b*d^3*e^3*(c + 1)*(8*a*c - 2*a + b*c) + 8*b*d ^3*e^3*(8*a + b)*(c - 1)*(c + 1)))/(128*d^2) + (x^3*(32*b*d^4*e^3*(8*a*c - 2*a + b*c) - 8*b*d^3*e^3*(d*(c - 1) + d*(c + 1))*(8*a + b) + 8*b*d^4*e^3* (8*a + b)*(c + 1)))/(192*d^2) - (x^3*(32*b^2*c*d^4*e^3 - 8*b^2*d^3*e^3*(d* (c - 1) + d*(c + 1)) + 8*b^2*d^4*e^3*(c - 1)))/(192*d^2) + (x*(((d*(c - 1) + d*(c + 1))*(((d*(c - 1) + d*(c + 1))*(32*b*d^4*e^3*(8*a*c - 2*a + b*c) - 8*b*d^3*e^3*(d*(c - 1) + d*(c + 1))*(8*a + b) + 8*b*d^4*e^3*(8*a + b)...