Integrand size = 20, antiderivative size = 236 \[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}+\frac {b^2 d x^2}{12 c}-\frac {b^2 d \text {arctanh}(c x)}{3 c^3}+\frac {b^2 d x \text {arctanh}(c x)}{2 c^2}+\frac {b d x^2 (a+b \text {arctanh}(c x))}{3 c}+\frac {1}{6} b d x^3 (a+b \text {arctanh}(c x))+\frac {d (a+b \text {arctanh}(c x))^2}{12 c^3}+\frac {1}{3} d x^3 (a+b \text {arctanh}(c x))^2+\frac {1}{4} c d x^4 (a+b \text {arctanh}(c x))^2-\frac {2 b d (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c^3}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^3}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3} \] Output:
1/2*a*b*d*x/c^2+1/3*b^2*d*x/c^2+1/12*b^2*d*x^2/c-1/3*b^2*d*arctanh(c*x)/c^ 3+1/2*b^2*d*x*arctanh(c*x)/c^2+1/3*b*d*x^2*(a+b*arctanh(c*x))/c+1/6*b*d*x^ 3*(a+b*arctanh(c*x))+1/12*d*(a+b*arctanh(c*x))^2/c^3+1/3*d*x^3*(a+b*arctan h(c*x))^2+1/4*c*d*x^4*(a+b*arctanh(c*x))^2-2/3*b*d*(a+b*arctanh(c*x))*ln(2 /(-c*x+1))/c^3+1/3*b^2*d*ln(-c^2*x^2+1)/c^3-1/3*b^2*d*polylog(2,1-2/(-c*x+ 1))/c^3
Time = 0.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.99 \[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {d \left (-b^2+6 a b c x+4 b^2 c x+4 a b c^2 x^2+b^2 c^2 x^2+4 a^2 c^3 x^3+2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 \left (-7+4 c^3 x^3+3 c^4 x^4\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (a c^3 x^3 (4+3 c x)+b \left (-2+3 c x+2 c^2 x^2+c^3 x^3\right )-4 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 a b \log (1-c x)-3 a b \log (1+c x)+4 b^2 \log \left (1-c^2 x^2\right )+4 a b \log \left (-1+c^2 x^2\right )+4 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{12 c^3} \] Input:
Integrate[x^2*(d + c*d*x)*(a + b*ArcTanh[c*x])^2,x]
Output:
(d*(-b^2 + 6*a*b*c*x + 4*b^2*c*x + 4*a*b*c^2*x^2 + b^2*c^2*x^2 + 4*a^2*c^3 *x^3 + 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(-7 + 4*c^3*x^3 + 3*c^4*x^4)*Ar cTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*c^3*x^3*(4 + 3*c*x) + b*(-2 + 3*c*x + 2 *c^2*x^2 + c^3*x^3) - 4*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 3*a*b*Log[1 - c* x] - 3*a*b*Log[1 + c*x] + 4*b^2*Log[1 - c^2*x^2] + 4*a*b*Log[-1 + c^2*x^2] + 4*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(12*c^3)
Time = 1.10 (sec) , antiderivative size = 236, 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.
| \(\displaystyle \int x^2 (c d x+d) (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c d x^3 (a+b \text {arctanh}(c x))^2+d x^2 (a+b \text {arctanh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d (a+b \text {arctanh}(c x))^2}{12 c^3}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{3 c^3}+\frac {1}{4} c d x^4 (a+b \text {arctanh}(c x))^2+\frac {1}{3} d x^3 (a+b \text {arctanh}(c x))^2+\frac {1}{6} b d x^3 (a+b \text {arctanh}(c x))+\frac {b d x^2 (a+b \text {arctanh}(c x))}{3 c}+\frac {a b d x}{2 c^2}-\frac {b^2 d \text {arctanh}(c x)}{3 c^3}+\frac {b^2 d x \text {arctanh}(c x)}{2 c^2}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3}+\frac {b^2 d x}{3 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^3}+\frac {b^2 d x^2}{12 c}\) |
Input:
Int[x^2*(d + c*d*x)*(a + b*ArcTanh[c*x])^2,x]
Output:
(a*b*d*x)/(2*c^2) + (b^2*d*x)/(3*c^2) + (b^2*d*x^2)/(12*c) - (b^2*d*ArcTan h[c*x])/(3*c^3) + (b^2*d*x*ArcTanh[c*x])/(2*c^2) + (b*d*x^2*(a + b*ArcTanh [c*x]))/(3*c) + (b*d*x^3*(a + b*ArcTanh[c*x]))/6 + (d*(a + b*ArcTanh[c*x]) ^2)/(12*c^3) + (d*x^3*(a + b*ArcTanh[c*x])^2)/3 + (c*d*x^4*(a + b*ArcTanh[ c*x])^2)/4 - (2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^3) + (b^2* d*Log[1 - c^2*x^2])/(3*c^3) - (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^3)
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 = 0.46 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.18
| method | result | size |
| parts | \(d \,a^{2} \left (\frac {1}{4} c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {d \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c x}{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{12}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{12}-\frac {\ln \left (c x +1\right )^{2}}{48}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{24}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right )^{2}}{48}-\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {c^{2} x^{2}}{12}+\frac {c x}{3}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{6}\right )}{c^{3}}+\frac {2 d a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{6}+\frac {c x}{4}+\frac {7 \ln \left (c x -1\right )}{24}+\frac {\ln \left (c x +1\right )}{24}\right )}{c^{3}}\) | \(278\) |
| derivativedivides | \(\frac {d \,a^{2} \left (\frac {1}{4} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+d \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c x}{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{12}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{12}-\frac {\ln \left (c x +1\right )^{2}}{48}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{24}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right )^{2}}{48}-\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {c^{2} x^{2}}{12}+\frac {c x}{3}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{6}\right )+2 d a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{6}+\frac {c x}{4}+\frac {7 \ln \left (c x -1\right )}{24}+\frac {\ln \left (c x +1\right )}{24}\right )}{c^{3}}\) | \(281\) |
| default | \(\frac {d \,a^{2} \left (\frac {1}{4} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+d \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{6}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c x}{2}+\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{12}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{12}-\frac {\ln \left (c x +1\right )^{2}}{48}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{24}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right )^{2}}{48}-\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {c^{2} x^{2}}{12}+\frac {c x}{3}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{6}\right )+2 d a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {x^{3} c^{3}}{12}+\frac {c^{2} x^{2}}{6}+\frac {c x}{4}+\frac {7 \ln \left (c x -1\right )}{24}+\frac {\ln \left (c x +1\right )}{24}\right )}{c^{3}}\) | \(281\) |
| risch | \(\frac {d \,b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}+\frac {d \,b^{2} \ln \left (-c x -1\right )}{6 c^{3}}+\frac {d b \,x^{3} a}{6}+\frac {d c \,x^{4} a^{2}}{4}+\frac {d \,b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}-\frac {d \,b^{2} \ln \left (-c x +1\right ) x^{3}}{12}-\frac {7 d \,b^{2} \ln \left (-c x +1\right )^{2}}{48 c^{3}}+\frac {d b \ln \left (-c x -1\right ) a}{12 c^{3}}+\left (-\frac {d \,b^{2} x^{3} \left (3 c x +4\right ) \ln \left (-c x +1\right )}{24}+\frac {d b \left (6 a \,c^{4} x^{4}+8 a \,c^{3} x^{3}+2 b \,c^{3} x^{3}+4 b \,c^{2} x^{2}+6 b c x +7 b \ln \left (-c x +1\right )\right )}{24 c^{3}}\right ) \ln \left (c x +1\right )-\frac {5 d \,b^{2}}{12 c^{3}}+\frac {d \,b^{2} \ln \left (-c x +1\right )}{2 c^{3}}-\frac {d b a}{c^{3}}-\frac {7 d \,a^{2}}{12 c^{3}}+\frac {d \,x^{3} a^{2}}{3}-\frac {d c a b \ln \left (-c x +1\right ) x^{4}}{4}-\frac {d \,b^{2} \ln \left (-c x +1\right ) x}{4 c^{2}}+\frac {d \,b^{2} \left (3 c^{4} x^{4}+4 x^{3} c^{3}+1\right ) \ln \left (c x +1\right )^{2}}{48 c^{3}}+\frac {d b \,x^{2} a}{3 c}-\frac {d a b \ln \left (-c x +1\right ) x^{3}}{3}+\frac {7 d a b \ln \left (-c x +1\right )}{12 c^{3}}+\frac {d c \,b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{16}-\frac {d \,b^{2} \ln \left (-c x +1\right ) x^{2}}{6 c}+\frac {a b d x}{2 c^{2}}+\frac {b^{2} d x}{3 c^{2}}+\frac {b^{2} d \,x^{2}}{12 c}-\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{3 c^{3}}+\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}\) | \(483\) |
Input:
int(x^2*(c*d*x+d)*(a+b*arctanh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
d*a^2*(1/4*c*x^4+1/3*x^3)+d*b^2/c^3*(1/4*arctanh(c*x)^2*c^4*x^4+1/3*arctan h(c*x)^2*c^3*x^3+1/6*arctanh(c*x)*c^3*x^3+1/3*arctanh(c*x)*c^2*x^2+1/2*arc tanh(c*x)*c*x+7/12*arctanh(c*x)*ln(c*x-1)+1/12*arctanh(c*x)*ln(c*x+1)-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)-1/3*dilog(1 /2*c*x+1/2)+7/48*ln(c*x-1)^2-7/24*ln(c*x-1)*ln(1/2*c*x+1/2)+1/12*c^2*x^2+1 /3*c*x+1/2*ln(c*x-1)+1/6*ln(c*x+1))+2*d*a*b/c^3*(1/4*arctanh(c*x)*c^4*x^4+ 1/3*arctanh(c*x)*c^3*x^3+1/12*x^3*c^3+1/6*c^2*x^2+1/4*c*x+7/24*ln(c*x-1)+1 /24*ln(c*x+1))
\[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(c*d*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="fricas")
Output:
integral(a^2*c*d*x^3 + a^2*d*x^2 + (b^2*c*d*x^3 + b^2*d*x^2)*arctanh(c*x)^ 2 + 2*(a*b*c*d*x^3 + a*b*d*x^2)*arctanh(c*x), x)
\[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=d \left (\int a^{2} x^{2}\, dx + \int a^{2} c x^{3}\, dx + \int b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int b^{2} c x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c x^{3} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \] Input:
integrate(x**2*(c*d*x+d)*(a+b*atanh(c*x))**2,x)
Output:
d*(Integral(a**2*x**2, x) + Integral(a**2*c*x**3, x) + Integral(b**2*x**2* atanh(c*x)**2, x) + Integral(2*a*b*x**2*atanh(c*x), x) + Integral(b**2*c*x **3*atanh(c*x)**2, x) + Integral(2*a*b*c*x**3*atanh(c*x), x))
Time = 0.28 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.70 \[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{4} \, a^{2} c d x^{4} + \frac {1}{3} \, a^{2} d x^{3} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a b c d + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b d + \frac {{\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d}{3 \, c^{3}} + \frac {b^{2} d \log \left (c x + 1\right )}{6 \, c^{3}} + \frac {b^{2} d \log \left (c x - 1\right )}{2 \, c^{3}} + \frac {4 \, b^{2} c^{2} d x^{2} + 16 \, b^{2} c d x + {\left (3 \, b^{2} c^{4} d x^{4} + 4 \, b^{2} c^{3} d x^{3} + b^{2} d\right )} \log \left (c x + 1\right )^{2} + {\left (3 \, b^{2} c^{4} d x^{4} + 4 \, b^{2} c^{3} d x^{3} - 7 \, b^{2} d\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (b^{2} c^{3} d x^{3} + 2 \, b^{2} c^{2} d x^{2} + 3 \, b^{2} c d x\right )} \log \left (c x + 1\right ) - 2 \, {\left (2 \, b^{2} c^{3} d x^{3} + 4 \, b^{2} c^{2} d x^{2} + 6 \, b^{2} c d x + {\left (3 \, b^{2} c^{4} d x^{4} + 4 \, b^{2} c^{3} d x^{3} + b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{48 \, c^{3}} \] Input:
integrate(x^2*(c*d*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="maxima")
Output:
1/4*a^2*c*d*x^4 + 1/3*a^2*d*x^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*c*d + 1/3*(2*x ^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*d + 1/3*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*d/c^3 + 1/6*b^2*d*lo g(c*x + 1)/c^3 + 1/2*b^2*d*log(c*x - 1)/c^3 + 1/48*(4*b^2*c^2*d*x^2 + 16*b ^2*c*d*x + (3*b^2*c^4*d*x^4 + 4*b^2*c^3*d*x^3 + b^2*d)*log(c*x + 1)^2 + (3 *b^2*c^4*d*x^4 + 4*b^2*c^3*d*x^3 - 7*b^2*d)*log(-c*x + 1)^2 + 4*(b^2*c^3*d *x^3 + 2*b^2*c^2*d*x^2 + 3*b^2*c*d*x)*log(c*x + 1) - 2*(2*b^2*c^3*d*x^3 + 4*b^2*c^2*d*x^2 + 6*b^2*c*d*x + (3*b^2*c^4*d*x^4 + 4*b^2*c^3*d*x^3 + b^2*d )*log(c*x + 1))*log(-c*x + 1))/c^3
\[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(c*d*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="giac")
Output:
integrate((c*d*x + d)*(b*arctanh(c*x) + a)^2*x^2, x)
Timed out. \[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \] Input:
int(x^2*(a + b*atanh(c*x))^2*(d + c*d*x),x)
Output:
int(x^2*(a + b*atanh(c*x))^2*(d + c*d*x), x)
\[ \int x^2 (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {d \left (3 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} x^{4}+4 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}-4 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -3 \mathit {atanh} \left (c x \right )^{2} b^{2}+6 \mathit {atanh} \left (c x \right ) a b \,c^{4} x^{4}+8 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}+2 \mathit {atanh} \left (c x \right ) a b +2 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}+4 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+6 \mathit {atanh} \left (c x \right ) b^{2} c x +4 \mathit {atanh} \left (c x \right ) b^{2}+4 \left (\int \mathit {atanh} \left (c x \right )^{2}d x \right ) b^{2} c +8 \,\mathrm {log}\left (c^{2} x -c \right ) a b +8 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2}+3 a^{2} c^{4} x^{4}+4 a^{2} c^{3} x^{3}+2 a b \,c^{3} x^{3}+4 a b \,c^{2} x^{2}+6 a b c x +b^{2} c^{2} x^{2}+4 b^{2} c x \right )}{12 c^{3}} \] Input:
int(x^2*(c*d*x+d)*(a+b*atanh(c*x))^2,x)
Output:
(d*(3*atanh(c*x)**2*b**2*c**4*x**4 + 4*atanh(c*x)**2*b**2*c**3*x**3 - 4*at anh(c*x)**2*b**2*c*x - 3*atanh(c*x)**2*b**2 + 6*atanh(c*x)*a*b*c**4*x**4 + 8*atanh(c*x)*a*b*c**3*x**3 + 2*atanh(c*x)*a*b + 2*atanh(c*x)*b**2*c**3*x* *3 + 4*atanh(c*x)*b**2*c**2*x**2 + 6*atanh(c*x)*b**2*c*x + 4*atanh(c*x)*b* *2 + 4*int(atanh(c*x)**2,x)*b**2*c + 8*log(c**2*x - c)*a*b + 8*log(c**2*x - c)*b**2 + 3*a**2*c**4*x**4 + 4*a**2*c**3*x**3 + 2*a*b*c**3*x**3 + 4*a*b* c**2*x**2 + 6*a*b*c*x + b**2*c**2*x**2 + 4*b**2*c*x))/(12*c**3)