Integrand size = 22, antiderivative size = 415 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {3 a b d^3 x}{2 c^3}+\frac {122 b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {44 b^2 d^3 x^3}{315 c}+\frac {1}{20} b^2 d^3 x^4+\frac {1}{105} b^2 c d^3 x^5-\frac {122 b^2 d^3 \text {arctanh}(c x)}{105 c^4}+\frac {3 b^2 d^3 x \text {arctanh}(c x)}{2 c^3}+\frac {26 b d^3 x^2 (a+b \text {arctanh}(c x))}{35 c^2}+\frac {b d^3 x^3 (a+b \text {arctanh}(c x))}{2 c}+\frac {13}{35} b d^3 x^4 (a+b \text {arctanh}(c x))+\frac {1}{5} b c d^3 x^5 (a+b \text {arctanh}(c x))+\frac {1}{21} b c^2 d^3 x^6 (a+b \text {arctanh}(c x))-\frac {d^3 (a+b \text {arctanh}(c x))^2}{140 c^4}+\frac {1}{4} d^3 x^4 (a+b \text {arctanh}(c x))^2+\frac {3}{5} c d^3 x^5 (a+b \text {arctanh}(c x))^2+\frac {1}{2} c^2 d^3 x^6 (a+b \text {arctanh}(c x))^2+\frac {1}{7} c^3 d^3 x^7 (a+b \text {arctanh}(c x))^2-\frac {52 b d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{35 c^4}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}-\frac {26 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{35 c^4} \]
3/2*a*b*d^3*x/c^3+122/105*b^2*d^3*x/c^3+7/20*b^2*d^3*x^2/c^2+44/315*b^2*d^ 3*x^3/c+1/20*b^2*d^3*x^4+1/105*b^2*c*d^3*x^5-122/105*b^2*d^3*arctanh(c*x)/ c^4+3/2*b^2*d^3*x*arctanh(c*x)/c^3+26/35*b*d^3*x^2*(a+b*arctanh(c*x))/c^2+ 1/2*b*d^3*x^3*(a+b*arctanh(c*x))/c+13/35*b*d^3*x^4*(a+b*arctanh(c*x))+1/5* b*c*d^3*x^5*(a+b*arctanh(c*x))+1/21*b*c^2*d^3*x^6*(a+b*arctanh(c*x))-1/140 *d^3*(a+b*arctanh(c*x))^2/c^4+1/4*d^3*x^4*(a+b*arctanh(c*x))^2+3/5*c*d^3*x ^5*(a+b*arctanh(c*x))^2+1/2*c^2*d^3*x^6*(a+b*arctanh(c*x))^2+1/7*c^3*d^3*x ^7*(a+b*arctanh(c*x))^2-52/35*b*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4+ 11/10*b^2*d^3*ln(-c^2*x^2+1)/c^4-26/35*b^2*d^3*polylog(2,1-2/(-c*x+1))/c^4
Time = 1.15 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.93 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^3 \left (-1464 a b-504 b^2+1890 a b c x+1464 b^2 c x+936 a b c^2 x^2+441 b^2 c^2 x^2+630 a b c^3 x^3+176 b^2 c^3 x^3+315 a^2 c^4 x^4+468 a b c^4 x^4+63 b^2 c^4 x^4+756 a^2 c^5 x^5+252 a b c^5 x^5+12 b^2 c^5 x^5+630 a^2 c^6 x^6+60 a b c^6 x^6+180 a^2 c^7 x^7+9 b^2 \left (-209+35 c^4 x^4+84 c^5 x^5+70 c^6 x^6+20 c^7 x^7\right ) \text {arctanh}(c x)^2+6 b \text {arctanh}(c x) \left (3 a c^4 x^4 \left (35+84 c x+70 c^2 x^2+20 c^3 x^3\right )+b \left (-244+315 c x+156 c^2 x^2+105 c^3 x^3+78 c^4 x^4+42 c^5 x^5+10 c^6 x^6\right )-312 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+945 a b \log (1-c x)-945 a b \log (1+c x)+1386 b^2 \log \left (1-c^2 x^2\right )+936 a b \log \left (-1+c^2 x^2\right )+936 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{1260 c^4} \]
(d^3*(-1464*a*b - 504*b^2 + 1890*a*b*c*x + 1464*b^2*c*x + 936*a*b*c^2*x^2 + 441*b^2*c^2*x^2 + 630*a*b*c^3*x^3 + 176*b^2*c^3*x^3 + 315*a^2*c^4*x^4 + 468*a*b*c^4*x^4 + 63*b^2*c^4*x^4 + 756*a^2*c^5*x^5 + 252*a*b*c^5*x^5 + 12* b^2*c^5*x^5 + 630*a^2*c^6*x^6 + 60*a*b*c^6*x^6 + 180*a^2*c^7*x^7 + 9*b^2*( -209 + 35*c^4*x^4 + 84*c^5*x^5 + 70*c^6*x^6 + 20*c^7*x^7)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c*x]*(3*a*c^4*x^4*(35 + 84*c*x + 70*c^2*x^2 + 20*c^3*x^3) + b *(-244 + 315*c*x + 156*c^2*x^2 + 105*c^3*x^3 + 78*c^4*x^4 + 42*c^5*x^5 + 1 0*c^6*x^6) - 312*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 945*a*b*Log[1 - c*x] - 945*a*b*Log[1 + c*x] + 1386*b^2*Log[1 - c^2*x^2] + 936*a*b*Log[-1 + c^2*x^ 2] + 936*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(1260*c^4)
Time = 1.59 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6502, 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.
| \(\displaystyle \int x^3 (c d x+d)^3 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c^3 d^3 x^6 (a+b \text {arctanh}(c x))^2+3 c^2 d^3 x^5 (a+b \text {arctanh}(c x))^2+3 c d^3 x^4 (a+b \text {arctanh}(c x))^2+d^3 x^3 (a+b \text {arctanh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 (a+b \text {arctanh}(c x))^2}{140 c^4}-\frac {52 b d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{35 c^4}+\frac {1}{7} c^3 d^3 x^7 (a+b \text {arctanh}(c x))^2+\frac {1}{2} c^2 d^3 x^6 (a+b \text {arctanh}(c x))^2+\frac {1}{21} b c^2 d^3 x^6 (a+b \text {arctanh}(c x))+\frac {26 b d^3 x^2 (a+b \text {arctanh}(c x))}{35 c^2}+\frac {3}{5} c d^3 x^5 (a+b \text {arctanh}(c x))^2+\frac {1}{5} b c d^3 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^3 x^4 (a+b \text {arctanh}(c x))^2+\frac {13}{35} b d^3 x^4 (a+b \text {arctanh}(c x))+\frac {b d^3 x^3 (a+b \text {arctanh}(c x))}{2 c}+\frac {3 a b d^3 x}{2 c^3}-\frac {122 b^2 d^3 \text {arctanh}(c x)}{105 c^4}+\frac {3 b^2 d^3 x \text {arctanh}(c x)}{2 c^3}-\frac {26 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{35 c^4}+\frac {122 b^2 d^3 x}{105 c^3}+\frac {7 b^2 d^3 x^2}{20 c^2}+\frac {11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}+\frac {1}{105} b^2 c d^3 x^5+\frac {44 b^2 d^3 x^3}{315 c}+\frac {1}{20} b^2 d^3 x^4\) |
(3*a*b*d^3*x)/(2*c^3) + (122*b^2*d^3*x)/(105*c^3) + (7*b^2*d^3*x^2)/(20*c^ 2) + (44*b^2*d^3*x^3)/(315*c) + (b^2*d^3*x^4)/20 + (b^2*c*d^3*x^5)/105 - ( 122*b^2*d^3*ArcTanh[c*x])/(105*c^4) + (3*b^2*d^3*x*ArcTanh[c*x])/(2*c^3) + (26*b*d^3*x^2*(a + b*ArcTanh[c*x]))/(35*c^2) + (b*d^3*x^3*(a + b*ArcTanh[ c*x]))/(2*c) + (13*b*d^3*x^4*(a + b*ArcTanh[c*x]))/35 + (b*c*d^3*x^5*(a + b*ArcTanh[c*x]))/5 + (b*c^2*d^3*x^6*(a + b*ArcTanh[c*x]))/21 - (d^3*(a + b *ArcTanh[c*x])^2)/(140*c^4) + (d^3*x^4*(a + b*ArcTanh[c*x])^2)/4 + (3*c*d^ 3*x^5*(a + b*ArcTanh[c*x])^2)/5 + (c^2*d^3*x^6*(a + b*ArcTanh[c*x])^2)/2 + (c^3*d^3*x^7*(a + b*ArcTanh[c*x])^2)/7 - (52*b*d^3*(a + b*ArcTanh[c*x])*L og[2/(1 - c*x)])/(35*c^4) + (11*b^2*d^3*Log[1 - c^2*x^2])/(10*c^4) - (26*b ^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/(35*c^4)
3.1.84.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 1.80 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.05
| method | result | size |
| parts | \(d^{3} a^{2} \left (\frac {1}{7} c^{3} x^{7}+\frac {1}{2} c^{2} x^{6}+\frac {3}{5} c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{2}+\frac {3 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{21}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {13 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{35}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{2}+\frac {26 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{35}+\frac {3 c x \,\operatorname {arctanh}\left (c x \right )}{2}+\frac {209 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{140}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{140}-\frac {26 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{35}-\frac {209 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{280}+\frac {209 \ln \left (c x -1\right )^{2}}{560}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{280}+\frac {\ln \left (c x +1\right )^{2}}{560}+\frac {c^{5} x^{5}}{105}+\frac {c^{4} x^{4}}{20}+\frac {44 c^{3} x^{3}}{315}+\frac {7 c^{2} x^{2}}{20}+\frac {122 c x}{105}+\frac {353 \ln \left (c x -1\right )}{210}+\frac {109 \ln \left (c x +1\right )}{210}\right )}{c^{4}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{2}+\frac {3 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+\frac {c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}+\frac {13 c^{4} x^{4}}{70}+\frac {c^{3} x^{3}}{4}+\frac {13 c^{2} x^{2}}{35}+\frac {3 c x}{4}+\frac {209 \ln \left (c x -1\right )}{280}-\frac {\ln \left (c x +1\right )}{280}\right )}{c^{4}}\) | \(436\) |
| derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{2} c^{6} x^{6}+\frac {3}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{2}+\frac {3 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{21}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {13 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{35}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{2}+\frac {26 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{35}+\frac {3 c x \,\operatorname {arctanh}\left (c x \right )}{2}+\frac {209 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{140}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{140}-\frac {26 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{35}-\frac {209 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{280}+\frac {209 \ln \left (c x -1\right )^{2}}{560}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{280}+\frac {\ln \left (c x +1\right )^{2}}{560}+\frac {c^{5} x^{5}}{105}+\frac {c^{4} x^{4}}{20}+\frac {44 c^{3} x^{3}}{315}+\frac {7 c^{2} x^{2}}{20}+\frac {122 c x}{105}+\frac {353 \ln \left (c x -1\right )}{210}+\frac {109 \ln \left (c x +1\right )}{210}\right )+2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{2}+\frac {3 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+\frac {c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}+\frac {13 c^{4} x^{4}}{70}+\frac {c^{3} x^{3}}{4}+\frac {13 c^{2} x^{2}}{35}+\frac {3 c x}{4}+\frac {209 \ln \left (c x -1\right )}{280}-\frac {\ln \left (c x +1\right )}{280}\right )}{c^{4}}\) | \(439\) |
| default | \(\frac {d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{2} c^{6} x^{6}+\frac {3}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{2}+\frac {3 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{21}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {13 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{35}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{2}+\frac {26 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{35}+\frac {3 c x \,\operatorname {arctanh}\left (c x \right )}{2}+\frac {209 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{140}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{140}-\frac {26 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{35}-\frac {209 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{280}+\frac {209 \ln \left (c x -1\right )^{2}}{560}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{280}+\frac {\ln \left (c x +1\right )^{2}}{560}+\frac {c^{5} x^{5}}{105}+\frac {c^{4} x^{4}}{20}+\frac {44 c^{3} x^{3}}{315}+\frac {7 c^{2} x^{2}}{20}+\frac {122 c x}{105}+\frac {353 \ln \left (c x -1\right )}{210}+\frac {109 \ln \left (c x +1\right )}{210}\right )+2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{2}+\frac {3 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+\frac {c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}+\frac {13 c^{4} x^{4}}{70}+\frac {c^{3} x^{3}}{4}+\frac {13 c^{2} x^{2}}{35}+\frac {3 c x}{4}+\frac {209 \ln \left (c x -1\right )}{280}-\frac {\ln \left (c x +1\right )}{280}\right )}{c^{4}}\) | \(439\) |
| risch | \(\frac {b^{2} d^{3} x^{4}}{20}+\frac {3 a b \,d^{3} x}{2 c^{3}}+\frac {122 b^{2} d^{3} x}{105 c^{3}}+\frac {7 b^{2} d^{3} x^{2}}{20 c^{2}}+\frac {44 b^{2} d^{3} x^{3}}{315 c}+\frac {b^{2} c \,d^{3} x^{5}}{105}-\frac {3 d^{3} c a b \ln \left (-c x +1\right ) x^{5}}{5}-\frac {d^{3} c^{2} a b \ln \left (-c x +1\right ) x^{6}}{2}-\frac {d^{3} c^{3} a b \ln \left (-c x +1\right ) x^{7}}{7}+\frac {353 d^{3} b^{2} \ln \left (-c x +1\right )}{210 c^{4}}+\frac {3 d^{3} c \,x^{5} a^{2}}{5}+\frac {d^{3} c^{2} x^{6} a^{2}}{2}+\frac {d^{3} c^{3} x^{7} a^{2}}{7}-\frac {209 d^{3} b^{2} \ln \left (-c x +1\right )^{2}}{560 c^{4}}+\frac {d^{3} b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{16}-\frac {209 d^{3} a^{2}}{140 c^{4}}+\frac {d^{3} x^{4} a^{2}}{4}+\frac {209 d^{3} b \ln \left (-c x +1\right ) a}{140 c^{4}}+\frac {3 d^{3} c \,b^{2} \ln \left (-c x +1\right )^{2} x^{5}}{20}+\frac {d^{3} c^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{6}}{8}+\frac {d^{3} c^{3} b^{2} \ln \left (-c x +1\right )^{2} x^{7}}{28}+\frac {d^{3} b^{2} \left (20 c^{7} x^{7}+70 c^{6} x^{6}+84 c^{5} x^{5}+35 c^{4} x^{4}-1\right ) \ln \left (c x +1\right )^{2}}{560 c^{4}}-\frac {d^{3} a b \ln \left (-c x +1\right ) x^{4}}{4}+\left (-\frac {d^{3} b^{2} x^{4} \left (20 c^{3} x^{3}+70 c^{2} x^{2}+84 c x +35\right ) \ln \left (-c x +1\right )}{280}-\frac {d^{3} b \left (-120 a \,c^{7} x^{7}-420 a \,c^{6} x^{6}-20 b \,c^{6} x^{6}-504 c^{5} x^{5} a -84 b \,c^{5} x^{5}-210 c^{4} x^{4} a -156 b \,c^{4} x^{4}-210 b \,c^{3} x^{3}-312 b \,c^{2} x^{2}-630 b c x -627 b \ln \left (-c x +1\right )\right )}{840 c^{4}}\right ) \ln \left (c x +1\right )+\frac {13 d^{3} b \,x^{4} a}{35}-\frac {13 d^{3} b^{2} \ln \left (-c x +1\right ) x^{4}}{70}+\frac {26 d^{3} b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{35 c^{4}}-\frac {353 d^{3} b a}{105 c^{4}}+\frac {109 d^{3} b^{2} \ln \left (-c x -1\right )}{210 c^{4}}-\frac {d^{3} b a \ln \left (-c x -1\right )}{140 c^{4}}-\frac {d^{3} b^{2} \ln \left (-c x +1\right ) x^{3}}{4 c}-\frac {13 d^{3} b^{2} \ln \left (-c x +1\right ) x^{2}}{35 c^{2}}-\frac {3 d^{3} b^{2} \ln \left (-c x +1\right ) x}{4 c^{3}}-\frac {26 d^{3} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{35 c^{4}}+\frac {26 d^{3} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{35 c^{4}}-\frac {d^{3} b^{2} c \ln \left (-c x +1\right ) x^{5}}{10}-\frac {d^{3} b^{2} c^{2} \ln \left (-c x +1\right ) x^{6}}{42}+\frac {26 d^{3} b a \,x^{2}}{35 c^{2}}+\frac {d^{3} b a \,x^{3}}{2 c}+\frac {d^{3} b c \,x^{5} a}{5}+\frac {d^{3} b \,c^{2} a \,x^{6}}{21}-\frac {77 d^{3} b^{2}}{45 c^{4}}\) | \(865\) |
d^3*a^2*(1/7*c^3*x^7+1/2*c^2*x^6+3/5*c*x^5+1/4*x^4)+d^3*b^2/c^4*(1/7*arcta nh(c*x)^2*c^7*x^7+1/2*c^6*x^6*arctanh(c*x)^2+3/5*c^5*x^5*arctanh(c*x)^2+1/ 4*c^4*x^4*arctanh(c*x)^2+1/21*c^6*x^6*arctanh(c*x)+1/5*c^5*x^5*arctanh(c*x )+13/35*c^4*x^4*arctanh(c*x)+1/2*c^3*x^3*arctanh(c*x)+26/35*c^2*x^2*arctan h(c*x)+3/2*c*x*arctanh(c*x)+209/140*arctanh(c*x)*ln(c*x-1)-1/140*arctanh(c *x)*ln(c*x+1)-26/35*dilog(1/2*c*x+1/2)-209/280*ln(c*x-1)*ln(1/2*c*x+1/2)+2 09/560*ln(c*x-1)^2-1/280*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/56 0*ln(c*x+1)^2+1/105*c^5*x^5+1/20*c^4*x^4+44/315*c^3*x^3+7/20*c^2*x^2+122/1 05*c*x+353/210*ln(c*x-1)+109/210*ln(c*x+1))+2*d^3*a*b/c^4*(1/7*arctanh(c*x )*c^7*x^7+1/2*c^6*x^6*arctanh(c*x)+3/5*c^5*x^5*arctanh(c*x)+1/4*c^4*x^4*ar ctanh(c*x)+1/42*c^6*x^6+1/10*c^5*x^5+13/70*c^4*x^4+1/4*c^3*x^3+13/35*c^2*x ^2+3/4*c*x+209/280*ln(c*x-1)-1/280*ln(c*x+1))
\[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
integral(a^2*c^3*d^3*x^6 + 3*a^2*c^2*d^3*x^5 + 3*a^2*c*d^3*x^4 + a^2*d^3*x ^3 + (b^2*c^3*d^3*x^6 + 3*b^2*c^2*d^3*x^5 + 3*b^2*c*d^3*x^4 + b^2*d^3*x^3) *arctanh(c*x)^2 + 2*(a*b*c^3*d^3*x^6 + 3*a*b*c^2*d^3*x^5 + 3*a*b*c*d^3*x^4 + a*b*d^3*x^3)*arctanh(c*x), x)
\[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=d^{3} \left (\int a^{2} x^{3}\, dx + \int 3 a^{2} c x^{4}\, dx + \int 3 a^{2} c^{2} x^{5}\, dx + \int a^{2} c^{3} x^{6}\, dx + \int b^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{5} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{6} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{4} \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{5} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{6} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**3*(Integral(a**2*x**3, x) + Integral(3*a**2*c*x**4, x) + Integral(3*a** 2*c**2*x**5, x) + Integral(a**2*c**3*x**6, x) + Integral(b**2*x**3*atanh(c *x)**2, x) + Integral(2*a*b*x**3*atanh(c*x), x) + Integral(3*b**2*c*x**4*a tanh(c*x)**2, x) + Integral(3*b**2*c**2*x**5*atanh(c*x)**2, x) + Integral( b**2*c**3*x**6*atanh(c*x)**2, x) + Integral(6*a*b*c*x**4*atanh(c*x), x) + Integral(6*a*b*c**2*x**5*atanh(c*x), x) + Integral(2*a*b*c**3*x**6*atanh(c *x), x))
Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (370) = 740\).
Time = 0.44 (sec) , antiderivative size = 928, normalized size of antiderivative = 2.24 \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\text {Too large to display} \]
1/7*a^2*c^3*d^3*x^7 + 1/2*a^2*c^2*d^3*x^6 + 3/5*a^2*c*d^3*x^5 + 1/4*b^2*d^ 3*x^4*arctanh(c*x)^2 + 1/42*(12*x^7*arctanh(c*x) + c*((2*c^4*x^6 + 3*c^2*x ^4 + 6*x^2)/c^6 + 6*log(c^2*x^2 - 1)/c^8))*a*b*c^3*d^3 + 1/4*a^2*d^3*x^4 + 1/30*(30*x^6*arctanh(c*x) + c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15* log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*a*b*c^2*d^3 + 3/10*(4*x^5*arctanh (c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*a*b*c*d^3 + 1/ 12*(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*b*d^3 + 1/48*(4*c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*arctanh(c*x) + (4*c^2*x^2 - 2*(3*log(c*x - 1) - 8)*log(c*x + 1) + 3*log(c*x + 1)^2 + 3*log(c*x - 1)^2 + 16*log(c*x - 1))/c^4)*b^2*d^3 + 26/35*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c* x + 1/2))*b^2*d^3/c^4 + 13/70*b^2*d^3*log(c*x + 1)/c^4 + 283/210*b^2*d^3*l og(c*x - 1)/c^4 + 1/2520*(24*b^2*c^5*d^3*x^5 + 126*b^2*c^4*d^3*x^4 + 352*b ^2*c^3*d^3*x^3 + 672*b^2*c^2*d^3*x^2 + 2928*b^2*c*d^3*x + 9*(10*b^2*c^7*d^ 3*x^7 + 35*b^2*c^6*d^3*x^6 + 42*b^2*c^5*d^3*x^5 + 17*b^2*d^3)*log(c*x + 1) ^2 + 9*(10*b^2*c^7*d^3*x^7 + 35*b^2*c^6*d^3*x^6 + 42*b^2*c^5*d^3*x^5 - 87* b^2*d^3)*log(-c*x + 1)^2 + 12*(5*b^2*c^6*d^3*x^6 + 21*b^2*c^5*d^3*x^5 + 39 *b^2*c^4*d^3*x^4 + 35*b^2*c^3*d^3*x^3 + 78*b^2*c^2*d^3*x^2 + 105*b^2*c*d^3 *x)*log(c*x + 1) - 6*(10*b^2*c^6*d^3*x^6 + 42*b^2*c^5*d^3*x^5 + 78*b^2*c^4 *d^3*x^4 + 70*b^2*c^3*d^3*x^3 + 156*b^2*c^2*d^3*x^2 + 210*b^2*c*d^3*x +...
\[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
Timed out. \[ \int x^3 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3 \,d x \]