Integrand size = 20, antiderivative size = 280 \[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {3 a b d^2 x}{2 c}+\frac {2 b^2 d^2 x}{3 c}+\frac {1}{12} b^2 d^2 x^2-\frac {2 b^2 d^2 \text {arctanh}(c x)}{3 c^2}+\frac {3 b^2 d^2 x \text {arctanh}(c x)}{2 c}+\frac {2}{3} b d^2 x^2 (a+b \text {arctanh}(c x))+\frac {1}{6} b c d^2 x^3 (a+b \text {arctanh}(c x))-\frac {d^2 (a+b \text {arctanh}(c x))^2}{12 c^2}+\frac {1}{2} d^2 x^2 (a+b \text {arctanh}(c x))^2+\frac {2}{3} c d^2 x^3 (a+b \text {arctanh}(c x))^2+\frac {1}{4} c^2 d^2 x^4 (a+b \text {arctanh}(c x))^2-\frac {4 b d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1-c^2 x^2\right )}{6 c^2}-\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^2} \] Output:
3/2*a*b*d^2*x/c+2/3*b^2*d^2*x/c+1/12*b^2*d^2*x^2-2/3*b^2*d^2*arctanh(c*x)/ c^2+3/2*b^2*d^2*x*arctanh(c*x)/c+2/3*b*d^2*x^2*(a+b*arctanh(c*x))+1/6*b*c* d^2*x^3*(a+b*arctanh(c*x))-1/12*d^2*(a+b*arctanh(c*x))^2/c^2+1/2*d^2*x^2*( a+b*arctanh(c*x))^2+2/3*c*d^2*x^3*(a+b*arctanh(c*x))^2+1/4*c^2*d^2*x^4*(a+ b*arctanh(c*x))^2-4/3*b*d^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^2+5/6*b^2* d^2*ln(-c^2*x^2+1)/c^2-2/3*b^2*d^2*polylog(2,1-2/(-c*x+1))/c^2
Time = 0.71 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.94 \[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^2 \left (-b^2+18 a b c x+8 b^2 c x+6 a^2 c^2 x^2+8 a b c^2 x^2+b^2 c^2 x^2+8 a^2 c^3 x^3+2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 \left (-17+6 c^2 x^2+8 c^3 x^3+3 c^4 x^4\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (a c^2 x^2 \left (6+8 c x+3 c^2 x^2\right )+b \left (-4+9 c x+4 c^2 x^2+c^3 x^3\right )-8 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+9 a b \log (1-c x)-9 a b \log (1+c x)+10 b^2 \log \left (1-c^2 x^2\right )+8 a b \log \left (-1+c^2 x^2\right )+8 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{12 c^2} \] Input:
Integrate[x*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2,x]
Output:
(d^2*(-b^2 + 18*a*b*c*x + 8*b^2*c*x + 6*a^2*c^2*x^2 + 8*a*b*c^2*x^2 + b^2* c^2*x^2 + 8*a^2*c^3*x^3 + 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(-17 + 6*c^2 *x^2 + 8*c^3*x^3 + 3*c^4*x^4)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*c^2*x^2 *(6 + 8*c*x + 3*c^2*x^2) + b*(-4 + 9*c*x + 4*c^2*x^2 + c^3*x^3) - 8*b*Log[ 1 + E^(-2*ArcTanh[c*x])]) + 9*a*b*Log[1 - c*x] - 9*a*b*Log[1 + c*x] + 10*b ^2*Log[1 - c^2*x^2] + 8*a*b*Log[-1 + c^2*x^2] + 8*b^2*PolyLog[2, -E^(-2*Ar cTanh[c*x])]))/(12*c^2)
Time = 1.23 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int x (c d x+d)^2 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c^2 d^2 x^3 (a+b \text {arctanh}(c x))^2+2 c d^2 x^2 (a+b \text {arctanh}(c x))^2+d^2 x (a+b \text {arctanh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} c^2 d^2 x^4 (a+b \text {arctanh}(c x))^2-\frac {d^2 (a+b \text {arctanh}(c x))^2}{12 c^2}-\frac {4 b d^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{3 c^2}+\frac {2}{3} c d^2 x^3 (a+b \text {arctanh}(c x))^2+\frac {1}{6} b c d^2 x^3 (a+b \text {arctanh}(c x))+\frac {1}{2} d^2 x^2 (a+b \text {arctanh}(c x))^2+\frac {2}{3} b d^2 x^2 (a+b \text {arctanh}(c x))+\frac {3 a b d^2 x}{2 c}-\frac {2 b^2 d^2 \text {arctanh}(c x)}{3 c^2}+\frac {3 b^2 d^2 x \text {arctanh}(c x)}{2 c}-\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1-c^2 x^2\right )}{6 c^2}+\frac {2 b^2 d^2 x}{3 c}+\frac {1}{12} b^2 d^2 x^2\) |
Input:
Int[x*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2,x]
Output:
(3*a*b*d^2*x)/(2*c) + (2*b^2*d^2*x)/(3*c) + (b^2*d^2*x^2)/12 - (2*b^2*d^2* ArcTanh[c*x])/(3*c^2) + (3*b^2*d^2*x*ArcTanh[c*x])/(2*c) + (2*b*d^2*x^2*(a + b*ArcTanh[c*x]))/3 + (b*c*d^2*x^3*(a + b*ArcTanh[c*x]))/6 - (d^2*(a + b *ArcTanh[c*x])^2)/(12*c^2) + (d^2*x^2*(a + b*ArcTanh[c*x])^2)/2 + (2*c*d^2 *x^3*(a + b*ArcTanh[c*x])^2)/3 + (c^2*d^2*x^4*(a + b*ArcTanh[c*x])^2)/4 - (4*b*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^2) + (5*b^2*d^2*Log[1 - c^2*x^2])/(6*c^2) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^2)
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 = 0.60 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.14
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{4} c^{2} x^{4}+\frac {2}{3} c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {2 \operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+\frac {17 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{12}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{12}+\frac {17 \ln \left (c x -1\right )^{2}}{48}-\frac {2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {17 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {\ln \left (c x +1\right )^{2}}{48}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {c^{2} x^{2}}{12}+\frac {2 c x}{3}+\frac {7 \ln \left (c x -1\right )}{6}+\frac {\ln \left (c x +1\right )}{2}\right )}{c^{2}}+\frac {2 d^{2} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{3}+\frac {3 c x}{4}+\frac {17 \ln \left (c x -1\right )}{24}-\frac {\ln \left (c x +1\right )}{24}\right )}{c^{2}}\) | \(318\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{4} c^{4} x^{4}+\frac {2}{3} x^{3} c^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {2 \operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+\frac {17 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{12}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{12}+\frac {17 \ln \left (c x -1\right )^{2}}{48}-\frac {2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {17 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {\ln \left (c x +1\right )^{2}}{48}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {c^{2} x^{2}}{12}+\frac {2 c x}{3}+\frac {7 \ln \left (c x -1\right )}{6}+\frac {\ln \left (c x +1\right )}{2}\right )+2 d^{2} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{3}+\frac {3 c x}{4}+\frac {17 \ln \left (c x -1\right )}{24}-\frac {\ln \left (c x +1\right )}{24}\right )}{c^{2}}\) | \(321\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{4} c^{4} x^{4}+\frac {2}{3} x^{3} c^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {2 \operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c x}{2}+\frac {17 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{12}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{12}+\frac {17 \ln \left (c x -1\right )^{2}}{48}-\frac {2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {17 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {\ln \left (c x +1\right )^{2}}{48}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {c^{2} x^{2}}{12}+\frac {2 c x}{3}+\frac {7 \ln \left (c x -1\right )}{6}+\frac {\ln \left (c x +1\right )}{2}\right )+2 d^{2} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{3}+\frac {3 c x}{4}+\frac {17 \ln \left (c x -1\right )}{24}-\frac {\ln \left (c x +1\right )}{24}\right )}{c^{2}}\) | \(321\) |
| risch | \(\frac {2 d^{2} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{2}}-\frac {2 d^{2} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{3 c^{2}}-\frac {d^{2} b \ln \left (-c x -1\right ) a}{12 c^{2}}+\frac {2 d^{2} b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{2}}+\frac {2 d^{2} b a \,x^{2}}{3}+\frac {d^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{2}}{8}+\frac {2 d^{2} c \,x^{3} a^{2}}{3}+\frac {d^{2} c^{2} x^{4} a^{2}}{4}-\frac {17 d^{2} b^{2} \ln \left (-c x +1\right )^{2}}{48 c^{2}}-\frac {d^{2} b^{2} \ln \left (-c x +1\right ) x^{2}}{3}-\frac {3 d^{2} b^{2}}{4 c^{2}}+\frac {7 d^{2} b^{2} \ln \left (-c x +1\right )}{6 c^{2}}-\frac {7 d^{2} b a}{3 c^{2}}-\frac {d^{2} c^{2} a b \ln \left (-c x +1\right ) x^{4}}{4}-\frac {2 d^{2} c a b \ln \left (-c x +1\right ) x^{3}}{3}+\frac {d^{2} b^{2} \ln \left (-c x -1\right )}{2 c^{2}}+\left (-\frac {d^{2} b^{2} x^{2} \left (3 c^{2} x^{2}+8 c x +6\right ) \ln \left (-c x +1\right )}{24}-\frac {d^{2} b \left (-6 a \,c^{4} x^{4}-16 a \,c^{3} x^{3}-2 b \,c^{3} x^{3}-12 a \,c^{2} x^{2}-8 b \,c^{2} x^{2}-18 b c x -17 b \ln \left (-c x +1\right )\right )}{24 c^{2}}\right ) \ln \left (c x +1\right )+\frac {2 b^{2} d^{2} x}{3 c}-\frac {17 d^{2} a^{2}}{12 c^{2}}+\frac {d^{2} x^{2} a^{2}}{2}+\frac {b^{2} d^{2} x^{2}}{12}+\frac {3 a b \,d^{2} x}{2 c}-\frac {d^{2} b^{2} c \ln \left (-c x +1\right ) x^{3}}{12}-\frac {3 d^{2} b^{2} \ln \left (-c x +1\right ) x}{4 c}+\frac {d^{2} c^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{16}+\frac {d^{2} c \,b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{6}+\frac {17 d^{2} a b \ln \left (-c x +1\right )}{12 c^{2}}-\frac {d^{2} a b \ln \left (-c x +1\right ) x^{2}}{2}+\frac {d^{2} b c a \,x^{3}}{6}+\frac {d^{2} b^{2} \left (3 c^{4} x^{4}+8 x^{3} c^{3}+6 c^{2} x^{2}-1\right ) \ln \left (c x +1\right )^{2}}{48 c^{2}}\) | \(614\) |
Input:
int(x*(c*d*x+d)^2*(a+b*arctanh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
d^2*a^2*(1/4*c^2*x^4+2/3*c*x^3+1/2*x^2)+d^2*b^2/c^2*(1/4*arctanh(c*x)^2*c^ 4*x^4+2/3*arctanh(c*x)^2*c^3*x^3+1/2*arctanh(c*x)^2*c^2*x^2+1/6*arctanh(c* x)*c^3*x^3+2/3*arctanh(c*x)*c^2*x^2+3/2*arctanh(c*x)*c*x+17/12*arctanh(c*x )*ln(c*x-1)-1/12*arctanh(c*x)*ln(c*x+1)+17/48*ln(c*x-1)^2-2/3*dilog(1/2*c* x+1/2)-17/24*ln(c*x-1)*ln(1/2*c*x+1/2)+1/48*ln(c*x+1)^2-1/24*(ln(c*x+1)-ln (1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/12*c^2*x^2+2/3*c*x+7/6*ln(c*x-1)+1/2*ln( c*x+1))+2*d^2*a*b/c^2*(1/4*arctanh(c*x)*c^4*x^4+2/3*arctanh(c*x)*c^3*x^3+1 /2*arctanh(c*x)*c^2*x^2+1/12*x^3*c^3+1/3*c^2*x^2+3/4*c*x+17/24*ln(c*x-1)-1 /24*ln(c*x+1))
\[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(c*d*x+d)^2*(a+b*arctanh(c*x))^2,x, algorithm="fricas")
Output:
integral(a^2*c^2*d^2*x^3 + 2*a^2*c*d^2*x^2 + a^2*d^2*x + (b^2*c^2*d^2*x^3 + 2*b^2*c*d^2*x^2 + b^2*d^2*x)*arctanh(c*x)^2 + 2*(a*b*c^2*d^2*x^3 + 2*a*b *c*d^2*x^2 + a*b*d^2*x)*arctanh(c*x), x)
\[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=d^{2} \left (\int a^{2} x\, dx + \int 2 a^{2} c x^{2}\, dx + \int a^{2} c^{2} x^{3}\, dx + \int b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 b^{2} c x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{3} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \] Input:
integrate(x*(c*d*x+d)**2*(a+b*atanh(c*x))**2,x)
Output:
d**2*(Integral(a**2*x, x) + Integral(2*a**2*c*x**2, x) + Integral(a**2*c** 2*x**3, x) + Integral(b**2*x*atanh(c*x)**2, x) + Integral(2*a*b*x*atanh(c* x), x) + Integral(2*b**2*c*x**2*atanh(c*x)**2, x) + Integral(b**2*c**2*x** 3*atanh(c*x)**2, x) + Integral(4*a*b*c*x**2*atanh(c*x), x) + Integral(2*a* b*c**2*x**3*atanh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (249) = 498\).
Time = 0.28 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.18 \[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x*(c*d*x+d)^2*(a+b*arctanh(c*x))^2,x, algorithm="maxima")
Output:
1/4*a^2*c^2*d^2*x^4 + 2/3*a^2*c*d^2*x^3 + 1/2*b^2*d^2*x^2*arctanh(c*x)^2 + 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*c^2*d^2 + 2/3*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*c*d^2 + 1/2*a^2*d^2*x^2 + 1/2*(2*x^2*arctanh( c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a*b*d^2 + 1/8*(4 *c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*arctanh(c*x) - (2*(log( c*x - 1) - 2)*log(c*x + 1) - log(c*x + 1)^2 - log(c*x - 1)^2 - 4*log(c*x - 1))/c^2)*b^2*d^2 + 2/3*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*d^2/c^2 + 2/3*b^2*d^2*log(c*x - 1)/c^2 + 1/48*(4*b^2*c^2*d^2*x ^2 + 32*b^2*c*d^2*x + (3*b^2*c^4*d^2*x^4 + 8*b^2*c^3*d^2*x^3 + 5*b^2*d^2)* log(c*x + 1)^2 + (3*b^2*c^4*d^2*x^4 + 8*b^2*c^3*d^2*x^3 - 11*b^2*d^2)*log( -c*x + 1)^2 + 4*(b^2*c^3*d^2*x^3 + 4*b^2*c^2*d^2*x^2 + 3*b^2*c*d^2*x)*log( c*x + 1) - 2*(2*b^2*c^3*d^2*x^3 + 8*b^2*c^2*d^2*x^2 + 6*b^2*c*d^2*x + (3*b ^2*c^4*d^2*x^4 + 8*b^2*c^3*d^2*x^3 + 5*b^2*d^2)*log(c*x + 1))*log(-c*x + 1 ))/c^2
Leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (249) = 498\).
Time = 1.45 (sec) , antiderivative size = 761, normalized size of antiderivative = 2.72 \[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x*(c*d*x+d)^2*(a+b*arctanh(c*x))^2,x, algorithm="giac")
Output:
2/45*(30*(c*x + 1)^3*b^2*d^2*log(-(c*x + 1)/(c*x - 1))^2/(((c*x + 1)^6*c^5 /(c*x - 1)^6 - 6*(c*x + 1)^5*c^5/(c*x - 1)^5 + 15*(c*x + 1)^4*c^5/(c*x - 1 )^4 - 20*(c*x + 1)^3*c^5/(c*x - 1)^3 + 15*(c*x + 1)^2*c^5/(c*x - 1)^2 - 6* (c*x + 1)*c^5/(c*x - 1) + c^5)*(c*x - 1)^3) + 2*(60*(c*x + 1)^3*a*b*d^2/(c *x - 1)^3 + 10*(c*x + 1)^3*b^2*d^2/(c*x - 1)^3 - 15*(c*x + 1)^2*b^2*d^2/(c *x - 1)^2 + 6*(c*x + 1)*b^2*d^2/(c*x - 1) - b^2*d^2)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^6*c^5/(c*x - 1)^6 - 6*(c*x + 1)^5*c^5/(c*x - 1)^5 + 15*(c* x + 1)^4*c^5/(c*x - 1)^4 - 20*(c*x + 1)^3*c^5/(c*x - 1)^3 + 15*(c*x + 1)^2 *c^5/(c*x - 1)^2 - 6*(c*x + 1)*c^5/(c*x - 1) + c^5) + (120*(c*x + 1)^3*a^2 *d^2/(c*x - 1)^3 + 40*(c*x + 1)^3*a*b*d^2/(c*x - 1)^3 - 60*(c*x + 1)^2*a*b *d^2/(c*x - 1)^2 + 24*(c*x + 1)*a*b*d^2/(c*x - 1) - 4*a*b*d^2 - 2*(c*x + 1 )^5*b^2*d^2/(c*x - 1)^5 + 11*(c*x + 1)^4*b^2*d^2/(c*x - 1)^4 - 18*(c*x + 1 )^3*b^2*d^2/(c*x - 1)^3 + 11*(c*x + 1)^2*b^2*d^2/(c*x - 1)^2 - 2*(c*x + 1) *b^2*d^2/(c*x - 1))/((c*x + 1)^6*c^5/(c*x - 1)^6 - 6*(c*x + 1)^5*c^5/(c*x - 1)^5 + 15*(c*x + 1)^4*c^5/(c*x - 1)^4 - 20*(c*x + 1)^3*c^5/(c*x - 1)^3 + 15*(c*x + 1)^2*c^5/(c*x - 1)^2 - 6*(c*x + 1)*c^5/(c*x - 1) + c^5) - 2*b^2 *d^2*log(-(c*x + 1)/(c*x - 1) + 1)/c^5 + 2*b^2*d^2*log(-(c*x + 1)/(c*x - 1 ))/c^5)*c^2
Timed out. \[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2 \,d x \] Input:
int(x*(a + b*atanh(c*x))^2*(d + c*d*x)^2,x)
Output:
int(x*(a + b*atanh(c*x))^2*(d + c*d*x)^2, x)
\[ \int x (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^{2} \left (3 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} x^{4}+8 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}+6 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-8 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -9 \mathit {atanh} \left (c x \right )^{2} b^{2}+6 \mathit {atanh} \left (c x \right ) a b \,c^{4} x^{4}+16 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}+12 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {atanh} \left (c x \right ) a b +2 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}+8 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+18 \mathit {atanh} \left (c x \right ) b^{2} c x +12 \mathit {atanh} \left (c x \right ) b^{2}+8 \left (\int \mathit {atanh} \left (c x \right )^{2}d x \right ) b^{2} c +16 \,\mathrm {log}\left (c^{2} x -c \right ) a b +20 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2}+3 a^{2} c^{4} x^{4}+8 a^{2} c^{3} x^{3}+6 a^{2} c^{2} x^{2}+2 a b \,c^{3} x^{3}+8 a b \,c^{2} x^{2}+18 a b c x +b^{2} c^{2} x^{2}+8 b^{2} c x \right )}{12 c^{2}} \] Input:
int(x*(c*d*x+d)^2*(a+b*atanh(c*x))^2,x)
Output:
(d**2*(3*atanh(c*x)**2*b**2*c**4*x**4 + 8*atanh(c*x)**2*b**2*c**3*x**3 + 6 *atanh(c*x)**2*b**2*c**2*x**2 - 8*atanh(c*x)**2*b**2*c*x - 9*atanh(c*x)**2 *b**2 + 6*atanh(c*x)*a*b*c**4*x**4 + 16*atanh(c*x)*a*b*c**3*x**3 + 12*atan h(c*x)*a*b*c**2*x**2 - 2*atanh(c*x)*a*b + 2*atanh(c*x)*b**2*c**3*x**3 + 8* atanh(c*x)*b**2*c**2*x**2 + 18*atanh(c*x)*b**2*c*x + 12*atanh(c*x)*b**2 + 8*int(atanh(c*x)**2,x)*b**2*c + 16*log(c**2*x - c)*a*b + 20*log(c**2*x - c )*b**2 + 3*a**2*c**4*x**4 + 8*a**2*c**3*x**3 + 6*a**2*c**2*x**2 + 2*a*b*c* *3*x**3 + 8*a*b*c**2*x**2 + 18*a*b*c*x + b**2*c**2*x**2 + 8*b**2*c*x))/(12 *c**2)