Integrand size = 18, antiderivative size = 496 \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\frac {b^2 c^3 d \arctan (c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \arctan (c x))^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}-\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2} \]
b^2*c^3*d*arctan(c*x)/(c^2*d^2+e^2)^2-b*c*(a+b*arctan(c*x))/(c^2*d^2+e^2)/ (e*x+d)+I*c^3*d*(a+b*arctan(c*x))^2/(c^2*d^2+e^2)^2+1/2*c^2*(c*d-e)*(c*d+e )*(a+b*arctan(c*x))^2/e/(c^2*d^2+e^2)^2-1/2*(a+b*arctan(c*x))^2/e/(e*x+d)^ 2-2*b*c^3*d*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/(c^2*d^2+e^2)^2+2*b*c^3*d*(a +b*arctan(c*x))*ln(2/(1+I*c*x))/(c^2*d^2+e^2)^2+b^2*c^2*e*ln(e*x+d)/(c^2*d ^2+e^2)^2+2*b*c^3*d*(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)^2-1/2*b^2*c^2*e*ln(c^2*x^2+1)/(c^2*d^2+e^2)^2+I*b^2*c^3*d*po lylog(2,1-2/(1-I*c*x))/(c^2*d^2+e^2)^2+I*b^2*c^3*d*polylog(2,1-2/(1+I*c*x) )/(c^2*d^2+e^2)^2-I*b^2*c^3*d*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x)) /(c^2*d^2+e^2)^2
Time = 6.54 (sec) , antiderivative size = 479, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=-\frac {a^2}{2 e (d+e x)^2}+\frac {a b \left (\left (-e^3+c^4 d^2 x (2 d+e x)-c^2 e \left (3 d^2+2 d e x+e^2 x^2\right )\right ) \arctan (c x)+c (d+e x) \left (-c^2 d^2-e^2+2 c^2 d (d+e x) \log (c (d+e x))-c^2 d (d+e x) \log \left (1+c^2 x^2\right )\right )\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}+\frac {b^2 c^2 \left (-\frac {2 e^{i \arctan \left (\frac {c d}{e}\right )} \arctan (c x)^2}{\sqrt {1+\frac {c^2 d^2}{e^2}} e}-\frac {e \left (1+c^2 x^2\right ) \arctan (c x)^2}{c^2 (d+e x)^2}+\frac {2 x \arctan (c x) (e+c d \arctan (c x))}{c d (d+e x)}+\frac {-2 e^2 \arctan (c x)+2 c d e \log \left (\frac {c (d+e x)}{\sqrt {1+c^2 x^2}}\right )}{c^3 d^3+c d e^2}-\frac {2 c d \left (-i \left (\pi -2 \arctan \left (\frac {c d}{e}\right )\right ) \arctan (c x)-\pi \log \left (1+e^{-2 i \arctan (c x)}\right )-2 \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )-\frac {1}{2} \pi \log \left (1+c^2 x^2\right )+2 \arctan \left (\frac {c d}{e}\right ) \log \left (\sin \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )\right )}{c^2 d^2+e^2}\right )}{2 \left (c^2 d^2+e^2\right )} \]
-1/2*a^2/(e*(d + e*x)^2) + (a*b*((-e^3 + c^4*d^2*x*(2*d + e*x) - c^2*e*(3* d^2 + 2*d*e*x + e^2*x^2))*ArcTan[c*x] + c*(d + e*x)*(-(c^2*d^2) - e^2 + 2* c^2*d*(d + e*x)*Log[c*(d + e*x)] - c^2*d*(d + e*x)*Log[1 + c^2*x^2])))/((c ^2*d^2 + e^2)^2*(d + e*x)^2) + (b^2*c^2*((-2*E^(I*ArcTan[(c*d)/e])*ArcTan[ c*x]^2)/(Sqrt[1 + (c^2*d^2)/e^2]*e) - (e*(1 + c^2*x^2)*ArcTan[c*x]^2)/(c^2 *(d + e*x)^2) + (2*x*ArcTan[c*x]*(e + c*d*ArcTan[c*x]))/(c*d*(d + e*x)) + (-2*e^2*ArcTan[c*x] + 2*c*d*e*Log[(c*(d + e*x))/Sqrt[1 + c^2*x^2]])/(c^3*d ^3 + c*d*e^2) - (2*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*Ar cTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*PolyLog[2, E^((2 *I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c^2*d^2 + e^2)))/(2*(c^2*d^2 + e^ 2))
Time = 0.81 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {b c \int \left (\frac {2 d e^2 (a+b \arctan (c x)) c^2}{\left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {\left (d^2 c^2-2 d e x c^2-e^2\right ) (a+b \arctan (c x)) c^2}{\left (c^2 d^2+e^2\right )^2 \left (c^2 x^2+1\right )}+\frac {e^2 (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)^2}\right )dx}{e}-\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \arctan (c x))^2}{2 e (d+e x)^2}+\frac {b c \left (\frac {i c^2 d e (a+b \arctan (c x))^2}{b \left (c^2 d^2+e^2\right )^2}+\frac {c (c d-e) (c d+e) (a+b \arctan (c x))^2}{2 b \left (c^2 d^2+e^2\right )^2}-\frac {e (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {2 c^2 d e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 c^2 d e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{\left (c^2 d^2+e^2\right )^2}+\frac {2 c^2 d e (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b c^2 d e \arctan (c x)}{\left (c^2 d^2+e^2\right )^2}+\frac {i b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {b c e^2 \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {b c e^2 \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}\right )}{e}\) |
-1/2*(a + b*ArcTan[c*x])^2/(e*(d + e*x)^2) + (b*c*((b*c^2*d*e*ArcTan[c*x]) /(c^2*d^2 + e^2)^2 - (e*(a + b*ArcTan[c*x]))/((c^2*d^2 + e^2)*(d + e*x)) + (I*c^2*d*e*(a + b*ArcTan[c*x])^2)/(b*(c^2*d^2 + e^2)^2) + (c*(c*d - e)*(c *d + e)*(a + b*ArcTan[c*x])^2)/(2*b*(c^2*d^2 + e^2)^2) - (2*c^2*d*e*(a + b *ArcTan[c*x])*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (2*c^2*d*e*(a + b*Ar cTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 + (b*c*e^2*Log[d + e*x])/ (c^2*d^2 + e^2)^2 + (2*c^2*d*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)^2 - (b*c*e^2*Log[1 + c^2*x^2])/(2 *(c^2*d^2 + e^2)^2) + (I*b*c^2*d*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (I*b*c^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 - (I*b*c^2*d*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/ (c^2*d^2 + e^2)^2))/e
3.1.14.3.1 Defintions of rubi rules used
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 = 28.13 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.47
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} c^{3}}{2 \left (c e x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\arctan \left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {\arctan \left (c x \right ) e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 \arctan \left (c x \right ) e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {\arctan \left (c x \right )^{2} c^{2} d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d e}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right )^{2} e^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e^{2} \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {e^{2} \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e d c \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e c d \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 )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {2 e^{2} c d \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 )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{e}\right )+2 a b \,c^{3} \left (-\frac {\arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{2 e}\right )}{c}\) | \(729\) |
| default | \(\frac {-\frac {a^{2} c^{3}}{2 \left (c e x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\arctan \left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {\arctan \left (c x \right ) e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 \arctan \left (c x \right ) e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {\arctan \left (c x \right )^{2} c^{2} d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d e}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right )^{2} e^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e^{2} \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {e^{2} \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e d c \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e c d \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 )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {2 e^{2} c d \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 )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{e}\right )+2 a b \,c^{3} \left (-\frac {\arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{2 e}\right )}{c}\) | \(729\) |
| parts | \(-\frac {a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {b^{2} \left (-\frac {c^{3} \arctan \left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (-\frac {\arctan \left (c x \right ) e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 \arctan \left (c x \right ) e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {\arctan \left (c x \right )^{2} c^{2} d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d e}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\arctan \left (c x \right )^{2} e^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e^{2} \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {e^{2} \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e d c \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {e c d \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 )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {2 c d \,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 )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}\right )}{e}\right )}{c}+\frac {2 a b \left (-\frac {c^{3} \arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (-\frac {e}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {2 e c d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {-c d e \ln \left (c^{2} x^{2}+1\right )+\left (c^{2} d^{2}-e^{2}\right ) \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}\right )}{2 e}\right )}{c}\) | \(731\) |
1/c*(-1/2*a^2*c^3/(c*e*x+c*d)^2/e+b^2*c^3*(-1/2/(c*e*x+c*d)^2/e*arctan(c*x )^2+1/e*(-arctan(c*x)*e/(c^2*d^2+e^2)/(c*e*x+c*d)+2*arctan(c*x)*e*c*d/(c^2 *d^2+e^2)^2*ln(c*e*x+c*d)+1/(c^2*d^2+e^2)^2*arctan(c*x)^2*c^2*d^2-arctan(c *x)/(c^2*d^2+e^2)^2*ln(c^2*x^2+1)*c*d*e-1/(c^2*d^2+e^2)^2*arctan(c*x)^2*e^ 2-1/2/(c^2*d^2+e^2)^2*(c^2*d^2-e^2)*arctan(c*x)^2+e^2/(c^2*d^2+e^2)^2*ln(c *e*x+c*d)-1/2*e^2/(c^2*d^2+e^2)^2*ln(c^2*x^2+1)+e/(c^2*d^2+e^2)^2*d*c*arct an(c*x)+e*c*d/(c^2*d^2+e^2)^2*(-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-dilog(-1/2* I*(c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/2*ln(c*x-I)^2)+1/2*I*(ln(c*x+I)* ln(c^2*x^2+1)-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))-1/2*ln(c*x+ I)^2))-2*e^2*c*d/(c^2*d^2+e^2)^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))-d ilog((I*e+c*e*x)/(I*e-c*d)))/e)))+2*a*b*c^3*(-1/2/(c*e*x+c*d)^2/e*arctan(c *x)+1/2/e*(-e/(c^2*d^2+e^2)/(c*e*x+c*d)+2*e*c*d/(c^2*d^2+e^2)^2*ln(c*e*x+c *d)+1/(c^2*d^2+e^2)^2*(-c*d*e*ln(c^2*x^2+1)+(c^2*d^2-e^2)*arctan(c*x)))))
\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
integral((b^2*arctan(c*x)^2 + 2*a*b*arctan(c*x) + a^2)/(e^3*x^3 + 3*d*e^2* x^2 + 3*d^2*e*x + d^3), x)
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]