\(\int \frac {a+b \text {arctanh}(c+d x)}{e+f x^2+g x^4} \, dx\) [78]

Optimal result

Integrand size = 25, antiderivative size = 1135 \[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^2+g x^4} \, dx =\text {Too large to display} \] Output:

-1/2*g^(1/2)*(a+b*arctanh(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)*(1-c)*g^(1/2)-d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/ 
(d*x+c+1))*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*arctanh(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)*(1-c)*g^(1/2)-d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+ 
1))*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*arctanh(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)*(1-c)*g^(1/2)+d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+1))*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*arct 
anh(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)*(1-c)*g^(1/2)+d*(-f+(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+1))*2^(1/2)/(-4* 
e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2))^(1/2)+1/4*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)*(1-c)*g^(1/2)- 
d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(d*x+c+1))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f 
-(-4*e*g+f^2)^(1/2))^(1/2)-1/4*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)*(1-c)*g^(1/2)-d*(-f+(-4*e*g+f^2) 
^(1/2))^(1/2))/(d*x+c+1))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f+(-4*e*g+f^2)^(1/2 
))^(1/2)-1/4*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)*(1-c)*g^(1/2)+d*(-f-(-4*e*g+f^2)^(1/2))^(1/2))/(d* 
x+c+1))*2^(1/2)/(-4*e*g+f^2)^(1/2)/(-f-(-4*e*g+f^2)^(1/2))^(1/2)+1/4*b*...
 

Mathematica [A] (warning: unable to verify)

Time = 3.21 (sec) , antiderivative size = 2111, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \text {arctanh}(c+d x)}{e+f x^2+g x^4} \, dx=\text {Result too large to show} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 4.67 (sec) , antiderivative size = 2120, 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.

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 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 {b \sqrt {g} \log (-c-d x+1) \log \left (-\frac {d \left (\sqrt {-f-\sqrt {f^2-4 e g}}-\sqrt {2} \sqrt {g} x\right )}{\sqrt {2} (1-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 {b \sqrt {g} \log (c+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+1)+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 {b \sqrt {g} \log (-c-d x+1) \log \left (-\frac {d \left (\sqrt {\sqrt {f^2-4 e g}-f}-\sqrt {2} \sqrt {g} x\right )}{\sqrt {2} (1-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 {b \sqrt {g} \log (c+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+1)+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 {b \sqrt {g} \log (c+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+1) \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 {b \sqrt {g} \log (-c-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} (1-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 {b \sqrt {g} \log (c+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+1) \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 {b \sqrt {g} \log (-c-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} (1-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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+1)}{\sqrt {2} (1-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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+1)}{\sqrt {2} \sqrt {g} (1-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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+1)}{\sqrt {2} (1-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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (-c-d x+1)}{\sqrt {2} \sqrt {g} (1-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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+1)}{\sqrt {2} (c+1) \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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+1)}{\sqrt {2} \sqrt {g} (c+1)+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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+1)}{\sqrt {2} (c+1) \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 {b \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sqrt {g} (c+d x+1)}{\sqrt {2} \sqrt {g} (c+1)+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*ArcTanh[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]]) + (b*Sqrt[g]*Log[1 - c - d*x]*Log[-((d*(Sqrt[- 
f - Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(1 - c)*Sqrt[g] - d* 
Sqrt[-f - Sqrt[f^2 - 4*e*g]]))])/(2*Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f - Sq 
rt[f^2 - 4*e*g]]) - (b*Sqrt[g]*Log[1 + c + d*x]*Log[(d*(Sqrt[-f - Sqrt[f^2 
 - 4*e*g]] - Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(1 + c)*Sqrt[g] + d*Sqrt[-f - Sq 
rt[f^2 - 4*e*g]])])/(2*Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f - Sqrt[f^2 - 4*e* 
g]]) - (b*Sqrt[g]*Log[1 - c - d*x]*Log[-((d*(Sqrt[-f + Sqrt[f^2 - 4*e*g]] 
- Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(1 - c)*Sqrt[g] - d*Sqrt[-f + Sqrt[f^2 - 4* 
e*g]]))])/(2*Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f + Sqrt[f^2 - 4*e*g]]) + (b* 
Sqrt[g]*Log[1 + c + d*x]*Log[(d*(Sqrt[-f + Sqrt[f^2 - 4*e*g]] - Sqrt[2]*Sq 
rt[g]*x))/(Sqrt[2]*(1 + c)*Sqrt[g] + d*Sqrt[-f + Sqrt[f^2 - 4*e*g]])])/(2* 
Sqrt[2]*Sqrt[f^2 - 4*e*g]*Sqrt[-f + Sqrt[f^2 - 4*e*g]]) + (b*Sqrt[g]*Log[1 
 + c + d*x]*Log[-((d*(Sqrt[-f - Sqrt[f^2 - 4*e*g]] + Sqrt[2]*Sqrt[g]*x))/( 
Sqrt[2]*(1 + c)*Sqrt[g] - d*Sqrt[-f - Sqrt[f^2 - 4*e*g]]))])/(2*Sqrt[2]*Sq 
rt[f^2 - 4*e*g]*Sqrt[-f - Sqrt[f^2 - 4*e*g]]) - (b*Sqrt[g]*Log[1 - c - d*x 
]*Log[(d*(Sqrt[-f - Sqrt[f^2 - 4*e*g]] + Sqrt[2]*Sqrt[g]*x))/(Sqrt[2]*(1 - 
 c)*Sqrt[g] + d*Sqrt[-f - Sqrt[f^2 - 4*e*g]])])/(2*Sqrt[2]*Sqrt[f^2 - 4...
 

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 = 1.54 (sec) , antiderivative size = 718, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {d^{3} a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{4}+\left (4 c g -4 g \right ) \textit {\_Z}^{3}+\left (6 c^{2} g +d^{2} f -12 c g +6 g \right ) \textit {\_Z}^{2}+\left (4 c^{3} g +2 c \,d^{2} f -12 c^{2} g -2 d^{2} f +12 c g -4 g \right ) \textit {\_Z} +c^{4} g +c^{2} d^{2} f +e \,d^{4}-4 c^{3} g -2 c \,d^{2} f +6 c^{2} g +d^{2} f -4 c g +g \right )}{\sum }\frac {\ln \left (-d x -\textit {\_R} -c +1\right )}{2 \textit {\_R}^{3} g +6 \textit {\_R}^{2} c g +6 \textit {\_R} \,c^{2} g +\textit {\_R} \,d^{2} f +2 c^{3} g +c \,d^{2} f -6 g \,\textit {\_R}^{2}-12 \textit {\_R} c g -6 c^{2} g -d^{2} f +6 g \textit {\_R} +6 c g -2 g}\right )}{2}+\frac {d^{3} b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{4}+\left (4 c g -4 g \right ) \textit {\_Z}^{3}+\left (6 c^{2} g +d^{2} f -12 c g +6 g \right ) \textit {\_Z}^{2}+\left (4 c^{3} g +2 c \,d^{2} f -12 c^{2} g -2 d^{2} f +12 c g -4 g \right ) \textit {\_Z} +c^{4} g +c^{2} d^{2} f +e \,d^{4}-4 c^{3} g -2 c \,d^{2} f +6 c^{2} g +d^{2} f -4 c g +g \right )}{\sum }\frac {\ln \left (-d x -c +1\right ) \ln \left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )}{2 \textit {\_R1}^{3} g +6 \textit {\_R1}^{2} c g +6 \textit {\_R1} \,c^{2} g +\textit {\_R1} \,d^{2} f +2 c^{3} g +c \,d^{2} f -6 \textit {\_R1}^{2} g -12 \textit {\_R1} c g -6 c^{2} g -d^{2} f +6 \textit {\_R1} g +6 c g -2 g}\right )}{4}+\frac {b \,d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{4}+\left (-4 c g -4 g \right ) \textit {\_Z}^{3}+\left (6 c^{2} g +d^{2} f +12 c g +6 g \right ) \textit {\_Z}^{2}+\left (-4 c^{3} g -2 c \,d^{2} f -12 c^{2} g -2 d^{2} f -12 c g -4 g \right ) \textit {\_Z} +c^{4} g +c^{2} d^{2} f +e \,d^{4}+4 c^{3} g +2 c \,d^{2} f +6 c^{2} g +d^{2} f +4 c g +g \right )}{\sum }\frac {\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )}{2 \textit {\_R1}^{3} g -6 \textit {\_R1}^{2} c g +6 \textit {\_R1} \,c^{2} g +\textit {\_R1} \,d^{2} f -2 c^{3} g -c \,d^{2} f -6 \textit {\_R1}^{2} g +12 \textit {\_R1} c g -6 c^{2} g -d^{2} f +6 \textit {\_R1} g -6 c g -2 g}\right )}{4}\)	\(718\)
parts	\(\text {Expression too large to display}\)	\(1740\)
derivativedivides	\(\text {Expression too large to display}\)	\(1746\)
default	\(\text {Expression too large to display}\)	\(1746\)

Input:

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

Output:

-1/2*d^3*a*sum(1/(2*_R^3*g+6*_R^2*c*g+6*_R*c^2*g+_R*d^2*f+2*c^3*g+c*d^2*f- 
6*_R^2*g-12*_R*c*g-6*c^2*g-d^2*f+6*_R*g+6*c*g-2*g)*ln(-d*x-_R-c+1),_R=Root 
Of(g*_Z^4+(4*c*g-4*g)*_Z^3+(6*c^2*g+d^2*f-12*c*g+6*g)*_Z^2+(4*c^3*g+2*c*d^ 
2*f-12*c^2*g-2*d^2*f+12*c*g-4*g)*_Z+c^4*g+c^2*d^2*f+e*d^4-4*c^3*g-2*c*d^2* 
f+6*c^2*g+d^2*f-4*c*g+g))+1/4*d^3*b*sum(1/(2*_R1^3*g+6*_R1^2*c*g+6*_R1*c^2 
*g+_R1*d^2*f+2*c^3*g+c*d^2*f-6*_R1^2*g-12*_R1*c*g-6*c^2*g-d^2*f+6*_R1*g+6* 
c*g-2*g)*(ln(-d*x-c+1)*ln((d*x+_R1+c-1)/_R1)+dilog((d*x+_R1+c-1)/_R1)),_R1 
=RootOf(g*_Z^4+(4*c*g-4*g)*_Z^3+(6*c^2*g+d^2*f-12*c*g+6*g)*_Z^2+(4*c^3*g+2 
*c*d^2*f-12*c^2*g-2*d^2*f+12*c*g-4*g)*_Z+c^4*g+c^2*d^2*f+e*d^4-4*c^3*g-2*c 
*d^2*f+6*c^2*g+d^2*f-4*c*g+g))+1/4*b*d^3*sum(1/(2*_R1^3*g-6*_R1^2*c*g+6*_R 
1*c^2*g+_R1*d^2*f-2*c^3*g-c*d^2*f-6*_R1^2*g+12*_R1*c*g-6*c^2*g-d^2*f+6*_R1 
*g-6*c*g-2*g)*(ln(d*x+c+1)*ln((-d*x+_R1-c-1)/_R1)+dilog((-d*x+_R1-c-1)/_R1 
)),_R1=RootOf(g*_Z^4+(-4*c*g-4*g)*_Z^3+(6*c^2*g+d^2*f+12*c*g+6*g)*_Z^2+(-4 
*c^3*g-2*c*d^2*f-12*c^2*g-2*d^2*f-12*c*g-4*g)*_Z+c^4*g+c^2*d^2*f+e*d^4+4*c 
^3*g+2*c*d^2*f+6*c^2*g+d^2*f+4*c*g+g))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

sage0*x
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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