Integrand size = 18, antiderivative size = 893 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=-\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 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 )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}} \]
-1/8*b*c/d/(c^2*d-e)/(e*x^2+d)+1/4*x*(a+b*arctan(c*x))/d/(e*x^2+d)^2+3/8*x *(a+b*arctan(c*x))/d^2/(e*x^2+d)-1/16*b*c*(5*c^2*d-3*e)*ln(c^2*x^2+1)/d^2/ (c^2*d-e)^2+1/16*b*c*(5*c^2*d-3*e)*ln(e*x^2+d)/d^2/(c^2*d-e)^2+3/8*(a+b*ar ctan(c*x))*arctan(x*e^(1/2)/d^(1/2))/d^(5/2)/e^(1/2)-3/32*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^(5/2)/(-c^2)^(1/2)/e^(1/2)+3/32*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^(5/2)/(-c ^2)^(1/2)/e^(1/2)-3/32*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^(5/2)/(-c^2)^(1/2)/e^(1/2) +3/32*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^(5/2)/(-c^2)^(1/2)/e^(1/2)+3/32*I*b*c*polylo g(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 ^(5/2)/(-c^2)^(1/2)/e^(1/2)-3/32*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^(5/2)/(-c^2)^(1/2)/e^(1/2)+3 /32*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^(5/2)/(-c^2)^(1/2)/e^(1/2)-3/32*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^(5/2)/(-c^2) ^(1/2)/e^(1/2)
Time = 10.77 (sec) , antiderivative size = 1745, normalized size of antiderivative = 1.95 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \]
(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqr t[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + (b*c*(10*c^2*d*Log[1 + ((c^2*d - e )*Cos[2*ArcTan[c*x]])/(c^2*d + e)] - 6*e*Log[1 + ((c^2*d - e)*Cos[2*ArcTan [c*x]])/(c^2*d + e)] + (3*c^2*d*(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)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^ 2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*C os[2*ArcTan[c*x]]])] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTan h[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(S qrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[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)*S qrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*S qrt[-(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)]*...
Time = 1.32 (sec) , antiderivative size = 875, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5447 |
| \(\displaystyle -b c \int \frac {\frac {3 e x^3+5 d x}{d^2 \left (e x^2+d\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \sqrt {e}}}{8 \left (c^2 x^2+1\right )}dx+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{8 d^{5/2} \sqrt {e}}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{8} b c \int \frac {\frac {3 e x^3+5 d x}{d^2 \left (e x^2+d\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \sqrt {e}}}{c^2 x^2+1}dx+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{8 d^{5/2} \sqrt {e}}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {1}{8} b c \int \left (\frac {x \left (3 e x^2+5 d\right )}{d^2 \left (c^2 x^2+1\right ) \left (e x^2+d\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \sqrt {e} \left (c^2 x^2+1\right )}\right )dx+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{8 d^{5/2} \sqrt {e}}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{8 d^{5/2} \sqrt {e}}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (e x^2+d\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (e x^2+d\right )^2}-\frac {1}{8} b c \left (-\frac {3 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^{5/2} \sqrt {e}}+\frac {3 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^{5/2} \sqrt {e}}+\frac {3 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^{5/2} \sqrt {e}}-\frac {3 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^{5/2} \sqrt {e}}+\frac {\left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right )}{2 d^2 \left (c^2 d-e\right )^2}-\frac {\left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right )}{2 d^2 \left (c^2 d-e\right )^2}-\frac {3 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^{5/2} \sqrt {e}}+\frac {3 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^{5/2} \sqrt {e}}-\frac {3 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^{5/2} \sqrt {e}}+\frac {3 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^{5/2} \sqrt {e}}+\frac {1}{d \left (c^2 d-e\right ) \left (e x^2+d\right )}\right )\) |
(x*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) + (3*x*(a + b*ArcTan[c*x]))/(8 *d^2*(d + e*x^2)) + (3*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8 *d^(5/2)*Sqrt[e]) - (b*c*(1/(d*(c^2*d - e)*(d + e*x^2)) - (((3*I)/4)*Log[( Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*S qrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (((3*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^(5/2)*Sqrt[e]) + (((3*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^(5/2)*Sqrt[e]) - (((3*I)/4)*Log[(Sqrt[e]*(1 + Sq rt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[ d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + ((5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(2* d^2*(c^2*d - e)^2) - ((5*c^2*d - 3*e)*Log[d + e*x^2])/(2*d^2*(c^2*d - e)^2 ) - (((3*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^(5/2)*Sqrt[e]) + (((3*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^(5/2)*Sqrt[e]) - (((3*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^(5/2)*Sq rt[e]) + (((3*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^(5/2)*Sqrt[e])))/8
3.12.71.3.1 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. 4006 vs. \(2 (681 ) = 1362\).
Time = 1.99 (sec) , antiderivative size = 4007, normalized size of antiderivative = 4.49
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(4007\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(4032\) |
| default | \(\text {Expression too large to display}\) | \(4032\) |
| risch | \(\text {Expression too large to display}\) | \(5059\) |
1/8*b*c^5/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e+1/4*a*x/d/(e*x^2+d )^2+3/8*a/d^2*x/(e*x^2+d)+3/8*a/d^2/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2))+1/ 8*b*c^7/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*x^2*e-3/16*b*c*(c^2*d* e)^(1/2)/d^2/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2* x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))-3/8*b*c*(c^2*d*e)^(1/2)/d^2/(c^4*d^2- 2*c^2*d*e+e^2)*arctan(c*x)^2+1/8*b*c^2*(e*d)^(1/2)/d^2*arctanh(1/4*(2*(c^2 *d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d *e+e^2)-3/16*b*(e*d)^(1/2)/d^3*e*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2 *x^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)+1/8*b*c^7/(c^4 *d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*x^4*e^2-9/8*b*c^3*arctan(c*x)^2/ d/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)*e-3/32*b*c^7*polylog(2,(c^2*d- e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/e/(c^4*d^2-2*c^2* d*e+e^2)^2*(c^2*d*e)^(1/2)*d+3/8*b*c*e^2*polylog(2,(c^2*d-e)*(1+I*c*x)^2/( c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/(c^4*d^2-2*c^2*d*e+e^2)^2/d^2*(c^ 2*d*e)^(1/2)-3/4*b*c/d^2/(c^4*d^2-2*c^2*d*e+e^2)*e^2/(c^2*d-e)*ln((1+I*c*x )/(c^2*x^2+1)^(1/2))+3/16*b*c/d^2/(c^4*d^2-2*c^2*d*e+e^2)*e^2/(c^2*d-e)*ln (c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e*(1+I*c* x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)-3/4*b*c^6/(c^4*d^2 -2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*arctan(c*x)*x^3*e^2+5/16*b*c^2*(e*d) ^(1/2)*e/d^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2...
\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]