Integrand size = 18, antiderivative size = 819 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx=\frac {x (a+b \arctan (c x))}{2 d \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}}-\frac {i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}}-\frac {i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}}+\frac {i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}}-\frac {b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right )}+\frac {b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right )}+\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}}-\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}}+\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}}-\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{3/2} \sqrt {e}} \]
1/2*x*(a+b*arctan(c*x))/d/(e*x^2+d)-1/4*b*c*ln(c^2*x^2+1)/d/(c^2*d-e)+1/4* b*c*ln(e*x^2+d)/d/(c^2*d-e)+1/2*(a+b*arctan(c*x))*arctan(x*e^(1/2)/d^(1/2) )/d^(3/2)/e^(1/2)-1/8*I*b*c*ln(-(1+x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2) *d^(1/2)-e^(1/2)))*ln(1-I*x*e^(1/2)/d^(1/2))/d^(3/2)/(-c^2)^(1/2)/e^(1/2)+ 1/8*I*b*c*ln((1-x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1/2)))* ln(1-I*x*e^(1/2)/d^(1/2))/d^(3/2)/(-c^2)^(1/2)/e^(1/2)-1/8*I*b*c*ln(-(1-x* (-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)-e^(1/2)))*ln(1+I*x*e^(1/2)/d ^(1/2))/d^(3/2)/(-c^2)^(1/2)/e^(1/2)+1/8*I*b*c*ln((1+x*(-c^2)^(1/2))*e^(1/ 2)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1/2)))*ln(1+I*x*e^(1/2)/d^(1/2))/d^(3/2)/(-c ^2)^(1/2)/e^(1/2)+1/8*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)-I*x*e^(1/2))/( (-c^2)^(1/2)*d^(1/2)-I*e^(1/2)))/d^(3/2)/(-c^2)^(1/2)/e^(1/2)-1/8*I*b*c*po lylog(2,(-c^2)^(1/2)*(d^(1/2)-I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)+I*e^(1/2) ))/d^(3/2)/(-c^2)^(1/2)/e^(1/2)+1/8*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)+ I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)-I*e^(1/2)))/d^(3/2)/(-c^2)^(1/2)/e^(1/2 )-1/8*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)+I*x*e^(1/2))/((-c^2)^(1/2)*d^( 1/2)+I*e^(1/2)))/d^(3/2)/(-c^2)^(1/2)/e^(1/2)
Time = 9.12 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx=\frac {a x}{2 d \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {b c \left (\frac {2 \log \left (1+\frac {\left (c^2 d-e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )}{c^2 d-e}+\frac {-4 \arctan (c x) \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )+2 \arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (-i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )\right )}{\sqrt {-c^2 d e}}+\frac {4 \arctan (c x) \sin (2 \arctan (c x))}{c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}\right )}{8 d} \]
(a*x)/(2*d*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[ e]) + (b*c*((2*Log[1 + ((c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(c^2 *d - e) + (-4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-(( c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[-((c^ 2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c ^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[ -(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c* e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/( (c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(c^2* d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sq rt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*Arc Tan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[-(( c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + Ar cTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTa n[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]] )] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[- (c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^ 2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))]))/Sqrt[-(c^2*d*e)] + (4*ArcTan[c *x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])))/...
Time = 1.17 (sec) , antiderivative size = 806, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5447, 27, 7276, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5447 |
| \(\displaystyle -b c \int \frac {\frac {x}{d \left (e x^2+d\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}}{2 \left (c^2 x^2+1\right )}dx+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 d^{3/2} \sqrt {e}}+\frac {x (a+b \arctan (c x))}{2 d \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} b c \int \frac {\frac {x}{d \left (e x^2+d\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}}{c^2 x^2+1}dx+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 d^{3/2} \sqrt {e}}+\frac {x (a+b \arctan (c x))}{2 d \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {1}{2} b c \int \left (\frac {x}{d \left (c^2 x^2+1\right ) \left (e x^2+d\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e} \left (c^2 x^2+1\right )}\right )dx+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 d^{3/2} \sqrt {e}}+\frac {x (a+b \arctan (c x))}{2 d \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 d^{3/2} \sqrt {e}}+\frac {x (a+b \arctan (c x))}{2 d \left (e x^2+d\right )}-\frac {1}{2} b c \left (-\frac {i \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}+\frac {i \log \left (-\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}+\frac {i \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}-\frac {i \log \left (\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}+\frac {\log \left (c^2 x^2+1\right )}{2 d \left (c^2 d-e\right )}-\frac {\log \left (e x^2+d\right )}{2 d \left (c^2 d-e\right )}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{4 \sqrt {-c^2} d^{3/2} \sqrt {e}}\right )\) |
(x*(a + b*ArcTan[c*x]))/(2*d*(d + e*x^2)) + ((a + b*ArcTan[c*x])*ArcTan[(S qrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]) - (b*c*(((-1/4*I)*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/S qrt[d]])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) + ((I/4)*Log[-((Sqrt[e]*(1 + Sqrt[-c ^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]]) /(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) + ((I/4)*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/ (I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[- c^2]*d^(3/2)*Sqrt[e]) - ((I/4)*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c ^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2 )*Sqrt[e]) + Log[1 + c^2*x^2]/(2*d*(c^2*d - e)) - Log[d + e*x^2]/(2*d*(c^2 *d - e)) - ((I/4)*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c ^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) + ((I/4)*PolyLog[2 , (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/ (Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - ((I/4)*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I* Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e] ) + ((I/4)*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqr t[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e])))/2
3.12.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ [{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2172 vs. \(2 (611 ) = 1222\).
Time = 1.16 (sec) , antiderivative size = 2173, normalized size of antiderivative = 2.65
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2173\) |
| parts | \(\text {Expression too large to display}\) | \(2305\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2320\) |
| default | \(\text {Expression too large to display}\) | \(2320\) |
1/8*c*b/d/(c^2*d-e)*ln((1-I*c*x)^2*e-c^2*d-2*(1-I*c*x)*e+e)-1/2*c^2*a/d/(- c^2*e*x^2-c^2*d)*x+1/4*c^2*b/(c^2*d-e)/(e*d)^(1/2)*arctanh(1/2*(2*(1-I*c*x )*e-2*e)/c/(e*d)^(1/2))+1/4*c^3*b*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d) +1/2*I*a/d/(e*d)^(1/2)*arctanh(1/2*(2*(1-I*c*x)*e-2*e)/c/(e*d)^(1/2))-1/8* c^4*b*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/ 2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*e*x^2+1/4*I*c^2*b*ln(1-I*c*x)/d/(c^2* d-e)/(-c^2*e*x^2-c^2*d)*e*x-1/8*b*c^4*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^ 2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*e*x^2 +1/8*b*c^4*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d )^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))*e*x^2-1/4*I*b*c^2*ln(1+I*c*x)/d/ (c^2*d-e)/(-c^2*e*x^2-c^2*d)*e*x+1/8*c^4*b*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x ^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1-I*c*x)*e-e)/(c*(e*d)^(1/2)-e))* e*x^2-1/8*c^2*b*ln(1-I*c*x)/d/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln( (c*(e*d)^(1/2)+(1-I*c*x)*e-e)/(c*(e*d)^(1/2)-e))*e^2*x^2+1/8*b*c^2*ln(1+I* c*x)/d/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x )*e+e)/(c*(e*d)^(1/2)+e))*e^2*x^2-1/8*b*c^2*ln(1+I*c*x)/d/(c^2*d-e)/(-c^2* e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e ))*e^2*x^2-1/4*I*c^4*b*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x+1/4*I*b* c^4*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x+1/8*c^2*b*ln(1-I*c*x)/d/(c^ 2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/...
\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]