\(\int \frac {a+b \arctan (c x)}{(d+e x^2)^3} \, dx\) [1170]

Optimal result

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}} \] Output:

-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)+3/8*(a+b*arctan(c*x))*arctan(e^(1/2)*x/d^ 
(1/2))/d^(5/2)/e^(1/2)+3/32*I*b*c*ln(e^(1/2)*(1-(-c^2)^(1/2)*x)/(I*(-c^2)^ 
(1/2)*d^(1/2)+e^(1/2)))*ln(1-I*e^(1/2)*x/d^(1/2))/(-c^2)^(1/2)/d^(5/2)/e^( 
1/2)-3/32*I*b*c*ln(-e^(1/2)*(1+(-c^2)^(1/2)*x)/(I*(-c^2)^(1/2)*d^(1/2)-e^( 
1/2)))*ln(1-I*e^(1/2)*x/d^(1/2))/(-c^2)^(1/2)/d^(5/2)/e^(1/2)-3/32*I*b*c*l 
n(-e^(1/2)*(1-(-c^2)^(1/2)*x)/(I*(-c^2)^(1/2)*d^(1/2)-e^(1/2)))*ln(1+I*e^( 
1/2)*x/d^(1/2))/(-c^2)^(1/2)/d^(5/2)/e^(1/2)+3/32*I*b*c*ln(e^(1/2)*(1+(-c^ 
2)^(1/2)*x)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1/2)))*ln(1+I*e^(1/2)*x/d^(1/2))/(- 
c^2)^(1/2)/d^(5/2)/e^(1/2)-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+3/32*I*b*c*polylo 
g(2,(-c^2)^(1/2)*(d^(1/2)-I*e^(1/2)*x)/((-c^2)^(1/2)*d^(1/2)-I*e^(1/2)))/( 
-c^2)^(1/2)/d^(5/2)/e^(1/2)-3/32*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)-I*e 
^(1/2)*x)/((-c^2)^(1/2)*d^(1/2)+I*e^(1/2)))/(-c^2)^(1/2)/d^(5/2)/e^(1/2)+3 
/32*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)+I*e^(1/2)*x)/((-c^2)^(1/2)*d^(1/ 
2)-I*e^(1/2)))/(-c^2)^(1/2)/d^(5/2)/e^(1/2)-3/32*I*b*c*polylog(2,(-c^2)^(1 
/2)*(d^(1/2)+I*e^(1/2)*x)/((-c^2)^(1/2)*d^(1/2)+I*e^(1/2)))/(-c^2)^(1/2)/d 
^(5/2)/e^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 10.39 (sec) , antiderivative size = 1745, normalized size of antiderivative = 1.95 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcTan[c*x])/(d + e*x^2)^3,x]
 

Output:

(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)]*...
 

Rubi [A] (verified)

Time = 1.39 (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 )\)

Input:

Int[(a + b*ArcTan[c*x])/(d + e*x^2)^3,x]
 

Output:

(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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5447

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])
 

rule 7276

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]
 

Maple [B] (verified)

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 = 3.43 (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	\(\text {Expression too large to display}\)	\(5059\)

Input:

int((a+b*arctan(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

3/8*b*c^4/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d^2*arctan(c*x)*x^3* 
e^3+5/8*b*c^4/(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* 
e^2+5/16*b*c^2*(d*e)^(1/2)/d^2*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/(d*e)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)-3/8* 
I*b*c*(c^2*d*e)^(1/2)/d^2/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)*ln(1-(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))+5/4*I*b*c^7/(c^4* 
d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*arctan(c*x)*x^2*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+3/4*b*c^5*arcta 
n(c*x)^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)+3/8*b*c^5*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*(c^2*d*e)^(1/2)+5/16*b*c^5/(c^4*d^2-2*c^2*d*e+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)-5/4*b*c^5/(c^4*d 
^2-2*c^2*d*e+e^2)/(c^2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-3/32*b*c^7*pol 
ylog(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*e^2*arctan(c*x)^2/(c^4*d^2-2*c^2*d*e+e^ 
2)^2/d^2*(c^2*d*e)^(1/2)-3/32*b/c*e^3*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^3*(c^...
 

Fricas [F]

\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate((a+b*arctan(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral((b*arctan(c*x) + a)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), 
x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((a+b*atan(c*x))/(e*x**2+d)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arctan(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
 

Output:

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
 

Giac [F]

\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate((a+b*arctan(c*x))/(e*x^2+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arctan(c*x) + a)/(e*x^2 + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((a + b*atan(c*x))/(d + e*x^2)^3,x)
 

Output:

int((a + b*atan(c*x))/(d + e*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\text {too large to display} \] Input:

int((a+b*atan(c*x))/(e*x^2+d)^3,x)
                                                                                    
                                                                                    
 

Output:

(9*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*c**8*d**6 + 18*sqrt(e)* 
sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*c**8*d**5*e*x**2 + 9*sqrt(e)*sqrt( 
d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*c**8*d**4*e**2*x**4 - 30*sqrt(e)*sqrt(d 
)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*c**4*d**4*e**2 - 60*sqrt(e)*sqrt(d)*atan 
((e*x)/(sqrt(e)*sqrt(d)))*a*c**4*d**3*e**3*x**2 - 30*sqrt(e)*sqrt(d)*atan( 
(e*x)/(sqrt(e)*sqrt(d)))*a*c**4*d**2*e**4*x**4 + 24*sqrt(e)*sqrt(d)*atan(( 
e*x)/(sqrt(e)*sqrt(d)))*a*c**2*d**3*e**3 + 48*sqrt(e)*sqrt(d)*atan((e*x)/( 
sqrt(e)*sqrt(d)))*a*c**2*d**2*e**4*x**2 + 24*sqrt(e)*sqrt(d)*atan((e*x)/(s 
qrt(e)*sqrt(d)))*a*c**2*d*e**5*x**4 - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e 
)*sqrt(d)))*a*d**2*e**4 - 6*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))* 
a*d*e**5*x**2 - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e**6*x** 
4 + 12*atan(c*x)**2*b*c**7*d**6*e + 24*atan(c*x)**2*b*c**7*d**5*e**2*x**2 
+ 12*atan(c*x)**2*b*c**7*d**4*e**3*x**4 - 24*atan(c*x)**2*b*c**5*d**5*e**2 
 - 48*atan(c*x)**2*b*c**5*d**4*e**3*x**2 - 24*atan(c*x)**2*b*c**5*d**3*e** 
4*x**4 + 12*atan(c*x)**2*b*c**3*d**4*e**3 + 24*atan(c*x)**2*b*c**3*d**3*e* 
*4*x**2 + 12*atan(c*x)**2*b*c**3*d**2*e**5*x**4 + 48*atan(c*x)*b*c**6*d**5 
*e**2*x + 24*atan(c*x)*b*c**6*d**4*e**3*x**3 - 104*atan(c*x)*b*c**4*d**4*e 
**3*x - 48*atan(c*x)*b*c**4*d**3*e**4*x**3 + 64*atan(c*x)*b*c**2*d**3*e**4 
*x + 24*atan(c*x)*b*c**2*d**2*e**5*x**3 - 8*atan(c*x)*b*d**2*e**5*x + 72*i 
nt((atan(c*x)*x**2)/(3*c**6*d**5*x**2 + 9*c**6*d**4*e*x**4 + 9*c**6*d**...