\(\int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx\) [67]

Optimal result

Integrand size = 25, antiderivative size = 1199 \[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx =\text {Too large to display} \] Output:

-1/2*g^(1/2)*(a+b*arctan(d*x+c))*ln(-2*d*((-f-(-4*e*g+f^2)^(1/2))^(1/2)-g^ 
(1/2)*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2)-d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/( 
1-I*(d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)+1/2 
*g^(1/2)*(a+b*arctan(d*x+c))*ln(-2*d*((-f+(-4*e*g+f^2)^(1/2))^(1/2)-g^(1/2 
)*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2)-d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(1-I* 
(d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)+1/2*g^( 
1/2)*(a+b*arctan(d*x+c))*ln(2*d*((-f-(-4*e*g+f^2)^(1/2))^(1/2)+g^(1/2)*x*2 
^(1/2))/(2^(1/2)*(I-c)*g^(1/2)+d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+ 
c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)-1/2*g^(1/2)* 
(a+b*arctan(d*x+c))*ln(2*d*((-f+(-4*e*g+f^2)^(1/2))^(1/2)+g^(1/2)*x*2^(1/2 
))/(2^(1/2)*(I-c)*g^(1/2)+d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))* 
2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)+1/4*I*b*g^(1/2)*p 
olylog(2,1+2*d*((-f-(-4*e*g+f^2)^(1/2))^(1/2)-g^(1/2)*x*2^(1/2))/(2^(1/2)* 
(I-c)*g^(1/2)-d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))*2^(1/2)/(-4* 
e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)-1/4*I*b*g^(1/2)*polylog(2,1+2 
*d*((-f+(-4*e*g+f^2)^(1/2))^(1/2)-g^(1/2)*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2 
)-d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/ 
2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)-1/4*I*b*g^(1/2)*polylog(2,1-2*d*((-f-(-4* 
e*g+f^2)^(1/2))^(1/2)+g^(1/2)*x*2^(1/2))/(2^(1/2)*(I-c)*g^(1/2)+d*(-f-(-4* 
e*g+f^2)^(1/2))^(1/2))/(1-I*(d*x+c)))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-...
 

Mathematica [A] (warning: unable to verify)

Time = 5.25 (sec) , antiderivative size = 1807, normalized size of antiderivative = 1.51 \[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcTan[c + d*x])/(e + f*x^2 + g*x^4),x]
 

Output:

(Sqrt[g]*((4*a*ArcTan[(Sqrt[2]*Sqrt[g]*x)/Sqrt[f - Sqrt[f^2 - 4*e*g]]])/Sq 
rt[f - Sqrt[f^2 - 4*e*g]] - (4*a*ArcTan[(Sqrt[2]*Sqrt[g]*x)/Sqrt[f + Sqrt[ 
f^2 - 4*e*g]]])/Sqrt[f + Sqrt[f^2 - 4*e*g]] + (I*b*Log[1 + I*c + I*d*x]*Lo 
g[(d*(Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(-I + c) 
*Sqrt[g] + d*Sqrt[-f - Sqrt[f^2 - 4*e*g]])])/Sqrt[-f - Sqrt[f^2 - 4*e*g]] 
- (I*b*Log[(-I)*(I + c + d*x)]*Log[(d*(Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt 
[2]*Sqrt[g]*x))/(Sqrt[2]*(I + c)*Sqrt[g] + d*Sqrt[-f - Sqrt[f^2 - 4*e*g]]) 
])/Sqrt[-f - Sqrt[f^2 - 4*e*g]] - (I*b*Log[1 + I*c + I*d*x]*Log[(d*(Sqrt[- 
f + Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(-I + c)*Sqrt[g] + d 
*Sqrt[-f + Sqrt[f^2 - 4*e*g]])])/Sqrt[-f + Sqrt[f^2 - 4*e*g]] + (I*b*Log[( 
-I)*(I + c + d*x)]*Log[(d*(Sqrt[-f + Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]* 
x))/(Sqrt[2]*(I + c)*Sqrt[g] + d*Sqrt[-f + Sqrt[f^2 - 4*e*g]])])/Sqrt[-f + 
 Sqrt[f^2 - 4*e*g]] - (I*b*Log[1 + I*c + I*d*x]*Log[(d*(Sqrt[-f - Sqrt[f^2 
 - 4*e*g]] + Sqrt[2]*Sqrt[g]*x))/(-(Sqrt[2]*(-I + c)*Sqrt[g]) + d*Sqrt[-f 
- Sqrt[f^2 - 4*e*g]])])/Sqrt[-f - Sqrt[f^2 - 4*e*g]] + (I*b*Log[(-I)*(I + 
c + d*x)]*Log[(d*(Sqrt[-f - Sqrt[f^2 - 4*e*g]] + Sqrt[2]*Sqrt[g]*x))/(-(Sq 
rt[2]*(I + c)*Sqrt[g]) + d*Sqrt[-f - Sqrt[f^2 - 4*e*g]])])/Sqrt[-f - Sqrt[ 
f^2 - 4*e*g]] + (I*b*Log[1 + I*c + I*d*x]*Log[(d*(Sqrt[-f + Sqrt[f^2 - 4*e 
*g]] + Sqrt[2]*Sqrt[g]*x))/(-(Sqrt[2]*(-I + c)*Sqrt[g]) + d*Sqrt[-f + Sqrt 
[f^2 - 4*e*g]])])/Sqrt[-f + Sqrt[f^2 - 4*e*g]] - (I*b*Log[(-I)*(I + c +...
 

Rubi [A] (warning: unable to verify)

Time = 4.78 (sec) , antiderivative size = 2244, normalized size of antiderivative = 1.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7279, 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.

\(\displaystyle \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {a}{e+f x^2+g x^4}+\frac {b \arctan (c+d x)}{e+f x^2+g x^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {2} a \sqrt {g} \arctan \left (\frac {\sqrt {2} \sqrt {g} x}{\sqrt {f-\sqrt {f^2-4 e g}}}\right )}{\sqrt {f^2-4 e g} \sqrt {f-\sqrt {f^2-4 e g}}}-\frac {\sqrt {2} a \sqrt {g} \arctan \left (\frac {\sqrt {2} \sqrt {g} x}{\sqrt {f+\sqrt {f^2-4 e g}}}\right )}{\sqrt {f^2-4 e g} \sqrt {f+\sqrt {f^2-4 e g}}}+\frac {i b \sqrt {g} \log (i c+i d x+1) \log \left (-\frac {d \left (\sqrt {-f-\sqrt {f^2-4 e g}}-\sqrt {2} \sqrt {g} x\right )}{\sqrt {2} (i-c) \sqrt {g}-d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \log (-i c-i d x+1) \log \left (\frac {d \left (\sqrt {-f-\sqrt {f^2-4 e g}}-\sqrt {2} \sqrt {g} x\right )}{\sqrt {2} \sqrt {g} (c+i)+d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \log (i c+i d x+1) \log \left (-\frac {d \left (\sqrt {\sqrt {f^2-4 e g}-f}-\sqrt {2} \sqrt {g} x\right )}{\sqrt {2} (i-c) \sqrt {g}-d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {i b \sqrt {g} \log (-i c-i d x+1) \log \left (\frac {d \left (\sqrt {\sqrt {f^2-4 e g}-f}-\sqrt {2} \sqrt {g} x\right )}{\sqrt {2} \sqrt {g} (c+i)+d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {i b \sqrt {g} \log (-i c-i d x+1) \log \left (-\frac {d \left (\sqrt {2} \sqrt {g} x+\sqrt {-f-\sqrt {f^2-4 e g}}\right )}{\sqrt {2} (c+i) \sqrt {g}-d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \log (i c+i d x+1) \log \left (\frac {d \left (\sqrt {2} \sqrt {g} x+\sqrt {-f-\sqrt {f^2-4 e g}}\right )}{\sqrt {2} \sqrt {g} (i-c)+d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \log (-i c-i d x+1) \log \left (-\frac {d \left (\sqrt {2} \sqrt {g} x+\sqrt {\sqrt {f^2-4 e g}-f}\right )}{\sqrt {2} (c+i) \sqrt {g}-d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {i b \sqrt {g} \log (i c+i d x+1) \log \left (\frac {d \left (\sqrt {2} \sqrt {g} x+\sqrt {\sqrt {f^2-4 e g}-f}\right )}{\sqrt {2} \sqrt {g} (i-c)+d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+i)}{\sqrt {2} (i-c) \sqrt {g}-d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+i)}{\sqrt {2} \sqrt {g} (i-c)+d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+i)}{\sqrt {2} (i-c) \sqrt {g}-d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+i)}{\sqrt {2} \sqrt {g} (i-c)+d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+i)}{\sqrt {2} (c+i) \sqrt {g}-d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+i)}{\sqrt {2} \sqrt {g} (c+i)+d \sqrt {-f-\sqrt {f^2-4 e g}}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {-f-\sqrt {f^2-4 e g}}}-\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+i)}{\sqrt {2} (c+i) \sqrt {g}-d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}+\frac {i b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+i)}{\sqrt {2} \sqrt {g} (c+i)+d \sqrt {\sqrt {f^2-4 e g}-f}}\right )}{2 \sqrt {2} \sqrt {f^2-4 e g} \sqrt {\sqrt {f^2-4 e g}-f}}\)

Input:

Int[(a + b*ArcTan[c + d*x])/(e + f*x^2 + g*x^4),x]
 

Output:

(Sqrt[2]*a*Sqrt[g]*ArcTan[(Sqrt[2]*Sqrt[g]*x)/Sqrt[f - Sqrt[f^2 - 4*e*g]]] 
)/(Sqrt[f^2 - 4*e*g]*Sqrt[f - Sqrt[f^2 - 4*e*g]]) - (Sqrt[2]*a*Sqrt[g]*Arc 
Tan[(Sqrt[2]*Sqrt[g]*x)/Sqrt[f + Sqrt[f^2 - 4*e*g]]])/(Sqrt[f^2 - 4*e*g]*S 
qrt[f + Sqrt[f^2 - 4*e*g]]) + ((I/2)*b*Sqrt[g]*Log[1 + I*c + I*d*x]*Log[-( 
(d*(Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(I - c)*Sq 
rt[g] - d*Sqrt[-f - Sqrt[f^2 - 4*e*g]]))])/(Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt 
[-f - Sqrt[f^2 - 4*e*g]]) - ((I/2)*b*Sqrt[g]*Log[1 - I*c - I*d*x]*Log[(d*( 
Sqrt[-f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(I + c)*Sqrt[g 
] + d*Sqrt[-f - Sqrt[f^2 - 4*e*g]])])/(Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f - 
 Sqrt[f^2 - 4*e*g]]) - ((I/2)*b*Sqrt[g]*Log[1 + I*c + I*d*x]*Log[-((d*(Sqr 
t[-f + Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(I - c)*Sqrt[g] - 
 d*Sqrt[-f + Sqrt[f^2 - 4*e*g]]))])/(Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f + S 
qrt[f^2 - 4*e*g]]) + ((I/2)*b*Sqrt[g]*Log[1 - I*c - I*d*x]*Log[(d*(Sqrt[-f 
 + Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(I + c)*Sqrt[g] + d*S 
qrt[-f + Sqrt[f^2 - 4*e*g]])])/(Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f + Sqrt[f 
^2 - 4*e*g]]) + ((I/2)*b*Sqrt[g]*Log[1 - I*c - I*d*x]*Log[-((d*(Sqrt[-f - 
Sqrt[f^2 - 4*e*g]] + Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(I + c)*Sqrt[g] - d*Sqrt 
[-f - Sqrt[f^2 - 4*e*g]]))])/(Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f - Sqrt[f^2 
 - 4*e*g]]) - ((I/2)*b*Sqrt[g]*Log[1 + I*c + I*d*x]*Log[(d*(Sqrt[-f - Sqrt 
[f^2 - 4*e*g]] + Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(I - c)*Sqrt[g] + d*Sqrt[...
 

Defintions of rubi rules used

rule 2009

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

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.03 (sec) , antiderivative size = 981, normalized size of antiderivative = 0.82

method	result	size
risch	\(\text {Expression too large to display}\)	\(981\)
parts	\(\text {Expression too large to display}\)	\(1831\)
derivativedivides	\(\text {Expression too large to display}\)	\(1838\)
default	\(\text {Expression too large to display}\)	\(1838\)

Input:

int((a+b*arctan(d*x+c))/(g*x^4+f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

-1/4*d^3*b*sum(1/(6*I*_R1^2*c*g-2*I*c^3*g-I*c*d^2*f-12*I*_R1*c*g+2*_R1^3*g 
-6*_R1*c^2*g-_R1*d^2*f+6*I*c*g-6*_R1^2*g+6*c^2*g+d^2*f+6*_R1*g-2*g)*(ln(1- 
I*c-I*d*x)*ln((_R1+I*d*x+I*c-1)/_R1)+dilog((_R1+I*d*x+I*c-1)/_R1)),_R1=Roo 
tOf(g*_Z^4+(4*RootOf(_Z^2+1,index=1)*c*g-4*g)*_Z^3+(-12*RootOf(_Z^2+1,inde 
x=1)*c*g-6*c^2*g-d^2*f+6*g)*_Z^2+(-4*RootOf(_Z^2+1,index=1)*c^3*g-2*RootOf 
(_Z^2+1,index=1)*c*d^2*f+12*RootOf(_Z^2+1,index=1)*c*g+12*c^2*g+2*d^2*f-4* 
g)*_Z+4*RootOf(_Z^2+1,index=1)*c^3*g+2*RootOf(_Z^2+1,index=1)*c*d^2*f+c^4* 
g+c^2*d^2*f+e*d^4-4*RootOf(_Z^2+1,index=1)*c*g-6*c^2*g-d^2*f+g))+1/2*I*d^3 
*a*sum(1/(6*I*_R^2*c*g-2*I*c^3*g-I*c*d^2*f-12*I*_R*c*g+2*_R^3*g-6*_R*c^2*g 
-_R*d^2*f+6*I*c*g-6*g*_R^2+6*c^2*g+d^2*f+6*g*_R-2*g)*ln(-I*d*x-I*c+1-_R),_ 
R=RootOf(g*_Z^4+(4*RootOf(_Z^2+1,index=1)*c*g-4*g)*_Z^3+(-12*RootOf(_Z^2+1 
,index=1)*c*g-6*c^2*g-d^2*f+6*g)*_Z^2+(-4*RootOf(_Z^2+1,index=1)*c^3*g-2*R 
ootOf(_Z^2+1,index=1)*c*d^2*f+12*RootOf(_Z^2+1,index=1)*c*g+12*c^2*g+2*d^2 
*f-4*g)*_Z+4*RootOf(_Z^2+1,index=1)*c^3*g+2*RootOf(_Z^2+1,index=1)*c*d^2*f 
+c^4*g+c^2*d^2*f+e*d^4-4*RootOf(_Z^2+1,index=1)*c*g-6*c^2*g-d^2*f+g))-1/4* 
b*d^3*sum(1/(-6*I*_R1^2*c*g+2*I*c^3*g+I*c*d^2*f+2*_R1^3*g-6*_R1*c^2*g+12*I 
*_R1*c*g-_R1*d^2*f-6*_R1^2*g+6*c^2*g-6*I*c*g+d^2*f+6*_R1*g-2*g)*(ln(1+I*c+ 
I*d*x)*ln((_R1-I*d*x-I*c-1)/_R1)+dilog((_R1-I*d*x-I*c-1)/_R1)),_R1=RootOf( 
g*_Z^4+(-4*RootOf(_Z^2+1,index=1)*c*g-4*g)*_Z^3+(-6*c^2*g+12*RootOf(_Z^2+1 
,index=1)*c*g-d^2*f+6*g)*_Z^2+(4*RootOf(_Z^2+1,index=1)*c^3*g+2*RootOf(...
 

Fricas [F]

\[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \arctan \left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:

integrate((a+b*arctan(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*atan(d*x+c))/(g*x**4+f*x**2+e),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \arctan \left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:

integrate((a+b*arctan(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="maxima")
 

Output:

integrate((b*arctan(d*x + c) + a)/(g*x^4 + f*x^2 + e), x)
 

Giac [F]

\[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx=\int { \frac {b \arctan \left (d x + c\right ) + a}{g x^{4} + f x^{2} + e} \,d x } \] Input:

integrate((a+b*arctan(d*x+c))/(g*x^4+f*x^2+e),x, algorithm="giac")
 

Output:

integrate((b*arctan(d*x + c) + a)/(g*x^4 + f*x^2 + e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c+d\,x\right )}{g\,x^4+f\,x^2+e} \,d x \] Input:

int((a + b*atan(c + d*x))/(e + f*x^2 + g*x^4),x)
 

Output:

int((a + b*atan(c + d*x))/(e + f*x^2 + g*x^4), x)
 

Reduce [F]

\[ \int \frac {a+b \arctan (c+d x)}{e+f x^2+g x^4} \, dx=\frac {2 \sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}-2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a f -4 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}-2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a e -2 \sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}+2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a f +4 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {g}\, \sqrt {e}-f}+2 \sqrt {g}\, x}{\sqrt {2 \sqrt {g}\, \sqrt {e}+f}}\right ) a e -\sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a f +\sqrt {e}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a f -2 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a e +2 \sqrt {g}\, \sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {g}\, \sqrt {e}-f}\, x +\sqrt {e}+\sqrt {g}\, x^{2}\right ) a e +16 \left (\int \frac {\mathit {atan} \left (d x +c \right )}{g \,x^{4}+f \,x^{2}+e}d x \right ) b \,e^{2} g -4 \left (\int \frac {\mathit {atan} \left (d x +c \right )}{g \,x^{4}+f \,x^{2}+e}d x \right ) b e \,f^{2}}{4 e \left (4 e g -f^{2}\right )} \] Input:

int((a+b*atan(d*x+c))/(g*x^4+f*x^2+e),x)
 

Output:

(2*sqrt(e)*sqrt(2*sqrt(g)*sqrt(e) + f)*atan((sqrt(2*sqrt(g)*sqrt(e) - f) - 
 2*sqrt(g)*x)/sqrt(2*sqrt(g)*sqrt(e) + f))*a*f - 4*sqrt(g)*sqrt(2*sqrt(g)* 
sqrt(e) + f)*atan((sqrt(2*sqrt(g)*sqrt(e) - f) - 2*sqrt(g)*x)/sqrt(2*sqrt( 
g)*sqrt(e) + f))*a*e - 2*sqrt(e)*sqrt(2*sqrt(g)*sqrt(e) + f)*atan((sqrt(2* 
sqrt(g)*sqrt(e) - f) + 2*sqrt(g)*x)/sqrt(2*sqrt(g)*sqrt(e) + f))*a*f + 4*s 
qrt(g)*sqrt(2*sqrt(g)*sqrt(e) + f)*atan((sqrt(2*sqrt(g)*sqrt(e) - f) + 2*s 
qrt(g)*x)/sqrt(2*sqrt(g)*sqrt(e) + f))*a*e - sqrt(e)*sqrt(2*sqrt(g)*sqrt(e 
) - f)*log( - sqrt(2*sqrt(g)*sqrt(e) - f)*x + sqrt(e) + sqrt(g)*x**2)*a*f 
+ sqrt(e)*sqrt(2*sqrt(g)*sqrt(e) - f)*log(sqrt(2*sqrt(g)*sqrt(e) - f)*x + 
sqrt(e) + sqrt(g)*x**2)*a*f - 2*sqrt(g)*sqrt(2*sqrt(g)*sqrt(e) - f)*log( - 
 sqrt(2*sqrt(g)*sqrt(e) - f)*x + sqrt(e) + sqrt(g)*x**2)*a*e + 2*sqrt(g)*s 
qrt(2*sqrt(g)*sqrt(e) - f)*log(sqrt(2*sqrt(g)*sqrt(e) - f)*x + sqrt(e) + s 
qrt(g)*x**2)*a*e + 16*int(atan(c + d*x)/(e + f*x**2 + g*x**4),x)*b*e**2*g 
- 4*int(atan(c + d*x)/(e + f*x**2 + g*x**4),x)*b*e*f**2)/(4*e*(4*e*g - f** 
2))