Integrand size = 18, antiderivative size = 321 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=-\frac {(a+b \text {arctanh}(c x))^2}{e (d+e x)}+\frac {b c (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{e (c d+e)}-\frac {b c (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{(c d-e) e}+\frac {2 b c (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{c^2 d^2-e^2}-\frac {2 b c (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{c^2 d^2-e^2}+\frac {b^2 c \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 e (c d+e)}+\frac {b^2 c \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 (c d-e) e}-\frac {b^2 c \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^2 d^2-e^2}+\frac {b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{c^2 d^2-e^2} \] Output:
-(a+b*arctanh(c*x))^2/e/(e*x+d)+b*c*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/e/(c *d+e)-b*c*(a+b*arctanh(c*x))*ln(2/(c*x+1))/(c*d-e)/e+2*b*c*(a+b*arctanh(c* x))*ln(2/(c*x+1))/(c^2*d^2-e^2)-2*b*c*(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c *d+e)/(c*x+1))/(c^2*d^2-e^2)+1/2*b^2*c*polylog(2,1-2/(-c*x+1))/e/(c*d+e)+1 /2*b^2*c*polylog(2,1-2/(c*x+1))/(c*d-e)/e-b^2*c*polylog(2,1-2/(c*x+1))/(c^ 2*d^2-e^2)+b^2*c*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c^2*d^2-e^2)
Result contains complex when optimal does not.
Time = 2.97 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=-\frac {a^2}{e (d+e x)}+\frac {a b c \left (-\frac {2 \text {arctanh}(c x)}{c d+c e x}+\frac {(-c d+e) \log (1-c x)+(c d+e) \log (1+c x)-2 e \log (c (d+e x))}{(c d-e) (c d+e)}\right )}{e}+\frac {b^2 \left (-\frac {e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2}{\sqrt {1-\frac {c^2 d^2}{e^2}} e}+\frac {x \text {arctanh}(c x)^2}{d+e x}+\frac {c d \left (i \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )-2 \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-i \pi \left (\text {arctanh}(c x)-\frac {1}{2} \log \left (1-c^2 x^2\right )\right )-2 \text {arctanh}\left (\frac {c d}{e}\right ) \left (\text {arctanh}(c x)+\log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-\log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{c^2 d^2-e^2}\right )}{d} \] Input:
Integrate[(a + b*ArcTanh[c*x])^2/(d + e*x)^2,x]
Output:
-(a^2/(e*(d + e*x))) + (a*b*c*((-2*ArcTanh[c*x])/(c*d + c*e*x) + ((-(c*d) + e)*Log[1 - c*x] + (c*d + e)*Log[1 + c*x] - 2*e*Log[c*(d + e*x)])/((c*d - e)*(c*d + e))))/e + (b^2*(-(ArcTanh[c*x]^2/(Sqrt[1 - (c^2*d^2)/e^2]*e*E^A rcTanh[(c*d)/e])) + (x*ArcTanh[c*x]^2)/(d + e*x) + (c*d*(I*Pi*Log[1 + E^(2 *ArcTanh[c*x])] - 2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh [c*x]))] - I*Pi*(ArcTanh[c*x] - Log[1 - c^2*x^2]/2) - 2*ArcTanh[(c*d)/e]*( ArcTanh[c*x] + Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - Log[I*S inh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(c*d)/e ] + ArcTanh[c*x]))]))/(c^2*d^2 - e^2)))/d
Time = 0.69 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.97, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {2 b c \int \left (-\frac {(a+b \text {arctanh}(c x)) e^2}{(c d-e) (c d+e) (d+e x)}+\frac {c (a+b \text {arctanh}(c x))}{2 (c d+e) (1-c x)}+\frac {c (a+b \text {arctanh}(c x))}{2 (c d-e) (c x+1)}\right )dx}{e}-\frac {(a+b \text {arctanh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b c \left (\frac {e \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^2 d^2-e^2}-\frac {e (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{c^2 d^2-e^2}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{2 (c d+e)}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{2 (c d-e)}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 (c d+e)}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{4 (c d-e)}\right )}{e}-\frac {(a+b \text {arctanh}(c x))^2}{e (d+e x)}\) |
Input:
Int[(a + b*ArcTanh[c*x])^2/(d + e*x)^2,x]
Output:
-((a + b*ArcTanh[c*x])^2/(e*(d + e*x))) + (2*b*c*(((a + b*ArcTanh[c*x])*Lo g[2/(1 - c*x)])/(2*(c*d + e)) - ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2 *(c*d - e)) + (e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^2*d^2 - e^2) - (e*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d ^2 - e^2) + (b*PolyLog[2, 1 - 2/(1 - c*x)])/(4*(c*d + e)) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)) - (b*e*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c^ 2*d^2 - e^2)) + (b*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))] )/(2*(c^2*d^2 - e^2))))/e
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]
Time = 0.41 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{\left (c e x +c d \right ) e}+\frac {-\frac {2 \,\operatorname {arctanh}\left (c x \right ) e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2 c d +2 e}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2 c d -2 e}-\frac {2 \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{\left (c d +e \right ) \left (c d -e \right )}+\frac {2 \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2 c d -2 e}-\frac {\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}}{c d +e}}{e}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {-\frac {e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {\ln \left (c x -1\right )}{2 c d +2 e}+\frac {\ln \left (c x +1\right )}{2 c d -2 e}}{e}\right )}{c}\) | \(451\) |
| default | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{\left (c e x +c d \right ) e}+\frac {-\frac {2 \,\operatorname {arctanh}\left (c x \right ) e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2 c d +2 e}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2 c d -2 e}-\frac {2 \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{\left (c d +e \right ) \left (c d -e \right )}+\frac {2 \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2 c d -2 e}-\frac {\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}}{c d +e}}{e}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {-\frac {e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {\ln \left (c x -1\right )}{2 c d +2 e}+\frac {\ln \left (c x +1\right )}{2 c d -2 e}}{e}\right )}{c}\) | \(451\) |
| parts | \(-\frac {a^{2}}{\left (e x +d \right ) e}+\frac {b^{2} \left (-\frac {c^{2} \operatorname {arctanh}\left (c x \right )^{2}}{\left (c e x +c d \right ) e}+\frac {2 c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2 c d +2 e}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2 c d -2 e}-\frac {\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{\left (c d +e \right ) \left (c d -e \right )}+\frac {-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}}{2 c d -2 e}-\frac {\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}}{2 \left (c d +e \right )}\right )}{e}\right )}{c}+\frac {2 a b \left (-\frac {c^{2} \operatorname {arctanh}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {c^{2} \left (-\frac {e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {\ln \left (c x -1\right )}{2 c d +2 e}+\frac {\ln \left (c x +1\right )}{2 c d -2 e}\right )}{e}\right )}{c}\) | \(453\) |
Input:
int((a+b*arctanh(c*x))^2/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/c*(-a^2*c^2/(c*e*x+c*d)/e+b^2*c^2*(-1/(c*e*x+c*d)/e*arctanh(c*x)^2+2/e*( -arctanh(c*x)*e/(c*d+e)/(c*d-e)*ln(c*e*x+c*d)-arctanh(c*x)/(2*c*d+2*e)*ln( c*x-1)+arctanh(c*x)/(2*c*d-2*e)*ln(c*x+1)-1/(c*d+e)/(c*d-e)*(1/2*e*(dilog( (c*e*x-e)/(-c*d-e))+ln(c*e*x+c*d)*ln((c*e*x-e)/(-c*d-e)))-1/2*e*(dilog((c* e*x+e)/(-c*d+e))+ln(c*e*x+c*d)*ln((c*e*x+e)/(-c*d+e))))+1/2/(c*d-e)*(-1/4* ln(c*x+1)^2+1/2*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)-1/2*dilog(1/2 *c*x+1/2))-1/2/(c*d+e)*(1/4*ln(c*x-1)^2-1/2*dilog(1/2*c*x+1/2)-1/2*ln(c*x- 1)*ln(1/2*c*x+1/2))))+2*a*b*c^2*(-1/(c*e*x+c*d)/e*arctanh(c*x)+1/e*(-e/(c* d+e)/(c*d-e)*ln(c*e*x+c*d)-1/(2*c*d+2*e)*ln(c*x-1)+1/(2*c*d-2*e)*ln(c*x+1) )))
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/(e*x+d)^2,x, algorithm="fricas")
Output:
integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(e^2*x^2 + 2*d*e* x + d^2), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((a+b*atanh(c*x))**2/(e*x+d)**2,x)
Output:
Integral((a + b*atanh(c*x))**2/(d + e*x)**2, x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/(e*x+d)^2,x, algorithm="maxima")
Output:
(c*(log(c*x + 1)/(c*d*e - e^2) - log(c*x - 1)/(c*d*e + e^2) - 2*log(e*x + d)/(c^2*d^2 - e^2)) - 2*arctanh(c*x)/(e^2*x + d*e))*a*b - 1/4*b^2*(log(-c* x + 1)^2/(e^2*x + d*e) + integrate(-((c*e*x - e)*log(c*x + 1)^2 + 2*(c*e*x + c*d - (c*e*x - e)*log(c*x + 1))*log(-c*x + 1))/(c*e^3*x^3 - d^2*e + (2* c*d*e^2 - e^3)*x^2 + (c*d^2*e - 2*d*e^2)*x), x)) - a^2/(e^2*x + d*e)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/(e*x+d)^2,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^2/(e*x + d)^2, x)
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((a + b*atanh(c*x))^2/(d + e*x)^2,x)
Output:
int((a + b*atanh(c*x))^2/(d + e*x)^2, x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^2} \, dx=\text {too large to display} \] Input:
int((a+b*atanh(c*x))^2/(e*x+d)^2,x)
Output:
(atanh(c*x)**2*b**2*c**4*d**4*x - atanh(c*x)**2*b**2*c**2*d**3*e - atanh(c *x)**2*b**2*c**2*d**2*e**2*x + atanh(c*x)**2*b**2*d*e**3 + 2*atanh(c*x)*a* b*c**4*d**4*x - 2*atanh(c*x)*a*b*e**4*x + 2*atanh(c*x)*b**2*c**3*d**3*e*x - 2*atanh(c*x)*b**2*c*d*e**3*x + 2*int((atanh(c*x)*x)/(c**4*d**4*x**2 + 2* c**4*d**3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d**4 - 2*c**2*d**3*e*x + 2*c **2*d*e**3*x**3 + c**2*e**4*x**4 - d**2*e**2 - 2*d*e**3*x - e**4*x**2),x)* b**2*c**7*d**8 + 2*int((atanh(c*x)*x)/(c**4*d**4*x**2 + 2*c**4*d**3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d**4 - 2*c**2*d**3*e*x + 2*c**2*d*e**3*x**3 + c**2*e**4*x**4 - d**2*e**2 - 2*d*e**3*x - e**4*x**2),x)*b**2*c**7*d**7*e *x - 2*int((atanh(c*x)*x)/(c**4*d**4*x**2 + 2*c**4*d**3*e*x**3 + c**4*d**2 *e**2*x**4 - c**2*d**4 - 2*c**2*d**3*e*x + 2*c**2*d*e**3*x**3 + c**2*e**4* x**4 - d**2*e**2 - 2*d*e**3*x - e**4*x**2),x)*b**2*c**5*d**6*e**2 - 2*int( (atanh(c*x)*x)/(c**4*d**4*x**2 + 2*c**4*d**3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d**4 - 2*c**2*d**3*e*x + 2*c**2*d*e**3*x**3 + c**2*e**4*x**4 - d**2 *e**2 - 2*d*e**3*x - e**4*x**2),x)*b**2*c**5*d**5*e**3*x - 2*int((atanh(c* x)*x)/(c**4*d**4*x**2 + 2*c**4*d**3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d* *4 - 2*c**2*d**3*e*x + 2*c**2*d*e**3*x**3 + c**2*e**4*x**4 - d**2*e**2 - 2 *d*e**3*x - e**4*x**2),x)*b**2*c**3*d**4*e**4 - 2*int((atanh(c*x)*x)/(c**4 *d**4*x**2 + 2*c**4*d**3*e*x**3 + c**4*d**2*e**2*x**4 - c**2*d**4 - 2*c**2 *d**3*e*x + 2*c**2*d*e**3*x**3 + c**2*e**4*x**4 - d**2*e**2 - 2*d*e**3*...