Integrand size = 18, antiderivative size = 341 \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx=\frac {i c (a+b \arctan (c x))^2}{c^2 d^2+e^2}+\frac {c^2 d (a+b \arctan (c x))^2}{e \left (c^2 d^2+e^2\right )}-\frac {(a+b \arctan (c x))^2}{e (d+e x)}-\frac {2 b c (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac {i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}-\frac {i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2} \] Output:
I*c*(a+b*arctan(c*x))^2/(c^2*d^2+e^2)+c^2*d*(a+b*arctan(c*x))^2/e/(c^2*d^2 +e^2)-(a+b*arctan(c*x))^2/e/(e*x+d)-2*b*c*(a+b*arctan(c*x))*ln(2/(1-I*c*x) )/(c^2*d^2+e^2)+2*b*c*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/(c^2*d^2+e^2)+2*b* c*(a+b*arctan(c*x))*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)+I*b^ 2*c*polylog(2,1-2/(1-I*c*x))/(c^2*d^2+e^2)+I*b^2*c*polylog(2,1-2/(1+I*c*x) )/(c^2*d^2+e^2)-I*b^2*c*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2* d^2+e^2)
Time = 1.87 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx=-\frac {a^2}{e (d+e x)}+\frac {a b \left (-2 \left (e-c^2 d x\right ) \arctan (c x)+c (d+e x) \left (2 \log (c (d+e x))-\log \left (1+c^2 x^2\right )\right )\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {b^2 \left (-\frac {e^{i \arctan \left (\frac {c d}{e}\right )} \arctan (c x)^2}{\sqrt {1+\frac {c^2 d^2}{e^2}} e}+\frac {x \arctan (c x)^2}{d+e x}-\frac {c d \left (-i \left (\pi -2 \arctan \left (\frac {c d}{e}\right )\right ) \arctan (c x)-\pi \log \left (1+e^{-2 i \arctan (c x)}\right )-2 \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )-\frac {1}{2} \pi \log \left (1+c^2 x^2\right )+2 \arctan \left (\frac {c d}{e}\right ) \log \left (\sin \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )\right )}{c^2 d^2+e^2}\right )}{d} \] Input:
Integrate[(a + b*ArcTan[c*x])^2/(d + e*x)^2,x]
Output:
-(a^2/(e*(d + e*x))) + (a*b*(-2*(e - c^2*d*x)*ArcTan[c*x] + c*(d + e*x)*(2 *Log[c*(d + e*x)] - Log[1 + c^2*x^2])))/((c^2*d^2 + e^2)*(d + e*x)) + (b^2 *(-((E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/(Sqrt[1 + (c^2*d^2)/e^2]*e)) + ( x*ArcTan[c*x]^2)/(d + e*x) - (c*d*((-I)*(Pi - 2*ArcTan[(c*d)/e])*ArcTan[c* x] - Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*(ArcTan[(c*d)/e] + ArcTan[c*x] )*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - (Pi*Log[1 + c^2*x^2 ])/2 + 2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*PolyL og[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c^2*d^2 + e^2)))/d
Time = 0.68 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5389, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {2 b c \int \left (\frac {(d-e x) (a+b \arctan (c x)) c^2}{\left (c^2 d^2+e^2\right ) \left (c^2 x^2+1\right )}+\frac {e^2 (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}\right )dx}{e}-\frac {(a+b \arctan (c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^2}{e (d+e x)}+\frac {2 b c \left (\frac {c d (a+b \arctan (c x))^2}{2 b \left (c^2 d^2+e^2\right )}+\frac {i e (a+b \arctan (c x))^2}{2 b \left (c^2 d^2+e^2\right )}-\frac {e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c^2 d^2+e^2}+\frac {e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^2 d^2+e^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 \left (c^2 d^2+e^2\right )}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}\right )}{e}\) |
Input:
Int[(a + b*ArcTan[c*x])^2/(d + e*x)^2,x]
Output:
-((a + b*ArcTan[c*x])^2/(e*(d + e*x))) + (2*b*c*((c*d*(a + b*ArcTan[c*x])^ 2)/(2*b*(c^2*d^2 + e^2)) + ((I/2)*e*(a + b*ArcTan[c*x])^2)/(b*(c^2*d^2 + e ^2)) - (e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2) + (e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2) + (e*(a + b*ArcTan[c* x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2) + ((I/ 2)*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2) + ((I/2)*b*e*PolyLog [2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2) - ((I/2)*b*e*PolyLog[2, 1 - (2*c*( d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)))/e
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S imp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTan[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 = 1.99 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.50
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{\left (c e x +c d \right ) e}+\frac {\frac {2 \arctan \left (c x \right ) e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c^{2} x^{2}+1\right )}{c^{2} d^{2}+e^{2}}+\frac {2 d c \arctan \left (c x \right )^{2}}{2 c^{2} d^{2}+2 e^{2}}-\frac {2 e^{2} \left (-\frac {i \ln \left (c e x +c d \right ) \left (\ln \left (\frac {-c e x +i e}{c d +i e}\right )-\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{c^{2} d^{2}+e^{2}}+\frac {2 e \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} d^{2}+2 e^{2}}}{e}\right )+2 a b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\frac {e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}+\frac {-\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \arctan \left (c x \right )}{c^{2} d^{2}+e^{2}}}{e}\right )}{c}\) | \(513\) |
| default | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\arctan \left (c x \right )^{2}}{\left (c e x +c d \right ) e}+\frac {\frac {2 \arctan \left (c x \right ) e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c^{2} x^{2}+1\right )}{c^{2} d^{2}+e^{2}}+\frac {2 d c \arctan \left (c x \right )^{2}}{2 c^{2} d^{2}+2 e^{2}}-\frac {2 e^{2} \left (-\frac {i \ln \left (c e x +c d \right ) \left (\ln \left (\frac {-c e x +i e}{c d +i e}\right )-\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{c^{2} d^{2}+e^{2}}+\frac {2 e \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} d^{2}+2 e^{2}}}{e}\right )+2 a b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\frac {e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}+\frac {-\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \arctan \left (c x \right )}{c^{2} d^{2}+e^{2}}}{e}\right )}{c}\) | \(513\) |
| parts | \(-\frac {a^{2}}{\left (e x +d \right ) e}+\frac {b^{2} \left (-\frac {c^{2} \arctan \left (c x \right )^{2}}{\left (c e x +c d \right ) e}+\frac {2 c^{2} \left (\frac {\arctan \left (c x \right ) e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )}+\frac {d c \arctan \left (c x \right )^{2}}{2 c^{2} d^{2}+2 e^{2}}-\frac {e^{2} \left (-\frac {i \ln \left (c e x +c d \right ) \left (\ln \left (\frac {-c e x +i e}{c d +i e}\right )-\ln \left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-c e x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {c e x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{c^{2} d^{2}+e^{2}}+\frac {e \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} d^{2}+2 e^{2}}\right )}{e}\right )}{c}+\frac {2 a b \left (-\frac {c^{2} \arctan \left (c x \right )}{\left (c e x +c d \right ) e}+\frac {c^{2} \left (\frac {e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}+\frac {-\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \arctan \left (c x \right )}{c^{2} d^{2}+e^{2}}\right )}{e}\right )}{c}\) | \(515\) |
Input:
int((a+b*arctan(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*arctan(c*x)^2+2/e*(a rctan(c*x)*e/(c^2*d^2+e^2)*ln(c*e*x+c*d)-1/2*arctan(c*x)/(c^2*d^2+e^2)*e*l n(c^2*x^2+1)+1/2/(c^2*d^2+e^2)*d*c*arctan(c*x)^2-e^2/(c^2*d^2+e^2)*(-1/2*I *ln(c*e*x+c*d)*(ln((I*e-c*e*x)/(c*d+I*e))-ln((I*e+c*e*x)/(I*e-c*d)))/e-1/2 *I*(dilog((I*e-c*e*x)/(c*d+I*e))-dilog((I*e+c*e*x)/(I*e-c*d)))/e)+1/2*e/(c ^2*d^2+e^2)*(-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I* (c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I)))+1/2*I*(ln(c*x+I)*ln(c^2*x^2+1)-1/2* ln(c*x+I)^2-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))))))+2*a*b*c^2 *(-1/(c*e*x+c*d)/e*arctan(c*x)+1/e*(e/(c^2*d^2+e^2)*ln(c*e*x+c*d)+1/(c^2*d ^2+e^2)*(-1/2*e*ln(c^2*x^2+1)+d*c*arctan(c*x)))))
\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arctan(c*x))^2/(e*x+d)^2,x, algorithm="fricas")
Output:
integral((b^2*arctan(c*x)^2 + 2*a*b*arctan(c*x) + a^2)/(e^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((a+b*atan(c*x))**2/(e*x+d)**2,x)
Output:
Integral((a + b*atan(c*x))**2/(d + e*x)**2, x)
\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arctan(c*x))^2/(e*x+d)^2,x, algorithm="maxima")
Output:
((2*c*d*arctan(c*x)/(c^2*d^2*e + e^3) - log(c^2*x^2 + 1)/(c^2*d^2 + e^2) + 2*log(e*x + d)/(c^2*d^2 + e^2))*c - 2*arctan(c*x)/(e^2*x + d*e))*a*b - 1/ 16*(4*arctan(c*x)^2 - 16*(e^2*x + d*e)*integrate(1/16*(12*(c^2*e*x^2 + e)* arctan(c*x)^2 + (c^2*e*x^2 + e)*log(c^2*x^2 + 1)^2 + 8*(c*e*x + c*d)*arcta n(c*x) - 4*(c^2*e*x^2 + c^2*d*x)*log(c^2*x^2 + 1))/(c^2*e^3*x^4 + 2*c^2*d* e^2*x^3 + 2*d*e^2*x + d^2*e + (c^2*d^2*e + e^3)*x^2), x) - log(c^2*x^2 + 1 )^2)*b^2/(e^2*x + d*e) - a^2/(e^2*x + d*e)
\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arctan(c*x))^2/(e*x+d)^2,x, algorithm="giac")
Output:
integrate((b*arctan(c*x) + a)^2/(e*x + d)^2, x)
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((a + b*atan(c*x))^2/(d + e*x)^2,x)
Output:
int((a + b*atan(c*x))^2/(d + e*x)^2, x)
\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
int((a+b*atan(c*x))^2/(e*x+d)^2,x)
Output:
(atan(c*x)**2*b**2*c**4*d**4*x + atan(c*x)**2*b**2*c**2*d**3*e + atan(c*x) **2*b**2*c**2*d**2*e**2*x + atan(c*x)**2*b**2*d*e**3 + 2*atan(c*x)*a*b*c** 4*d**4*x - 2*atan(c*x)*a*b*c**2*d**3*e - 2*atan(c*x)*a*b*c**2*d**2*e**2*x + 2*atan(c*x)*a*b*d*e**3 - 2*atan(c*x)*b**2*c**3*d**3*e*x + 2*atan(c*x)*b* *2*c*d**2*e**2 - 2*int((atan(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((atan(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((atan(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((atan(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((atan(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((atan(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*...