\(\int \frac {1}{x^2 (a+b+2 a x^2+a x^4)} \, dx\) [843]

Optimal result

Integrand size = 20, antiderivative size = 328 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=-\frac {1}{(a+b) x}+\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} (a+b)^{3/2} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} (a+b)^{3/2} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {\sqrt [4]{a} \left (2 \sqrt {a}-\sqrt {a+b}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt {a+b}+\sqrt {a} x^2}\right )}{2 \sqrt {2} (a+b)^{3/2} \sqrt {-\sqrt {a}+\sqrt {a+b}}} \] Output:

-1/(a+b)/x+1/4*a^(1/4)*(2*a^(1/2)+(a+b)^(1/2))*arctan(((-a^(1/2)+(a+b)^(1/ 
2))^(1/2)-2^(1/2)*a^(1/4)*x)/(a^(1/2)+(a+b)^(1/2))^(1/2))*2^(1/2)/(a+b)^(3 
/2)/(a^(1/2)+(a+b)^(1/2))^(1/2)-1/4*a^(1/4)*(2*a^(1/2)+(a+b)^(1/2))*arctan 
(((-a^(1/2)+(a+b)^(1/2))^(1/2)+2^(1/2)*a^(1/4)*x)/(a^(1/2)+(a+b)^(1/2))^(1 
/2))*2^(1/2)/(a+b)^(3/2)/(a^(1/2)+(a+b)^(1/2))^(1/2)-1/4*a^(1/4)*(2*a^(1/2 
)-(a+b)^(1/2))*arctanh(2^(1/2)*a^(1/4)*(-a^(1/2)+(a+b)^(1/2))^(1/2)*x/((a+ 
b)^(1/2)+a^(1/2)*x^2))*2^(1/2)/(a+b)^(3/2)/(-a^(1/2)+(a+b)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\frac {1}{(-a-b) x}+\frac {\left (i a-\sqrt {a} \sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a-i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a-i \sqrt {a} \sqrt {b}} \sqrt {b} (a+b)}+\frac {\left (-i a-\sqrt {a} \sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a+i \sqrt {a} \sqrt {b}} \sqrt {b} (a+b)} \] Input:

Integrate[1/(x^2*(a + b + 2*a*x^2 + a*x^4)),x]
 

Output:

1/((-a - b)*x) + ((I*a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sq 
rt[a]*Sqrt[b]]])/(2*Sqrt[a - I*Sqrt[a]*Sqrt[b]]*Sqrt[b]*(a + b)) + (((-I)* 
a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(2*S 
qrt[a + I*Sqrt[a]*Sqrt[b]]*Sqrt[b]*(a + b))
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {1443, 25, 27, 1483, 27, 1142, 25, 27, 1083, 217, 1103}

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.

\(\displaystyle \int \frac {1}{x^2 \left (a x^4+2 a x^2+a+b\right )} \, dx\)

\(\Big \downarrow \) 1443

\(\displaystyle \frac {\int -\frac {a \left (x^2+2\right )}{a x^4+2 a x^2+a+b}dx}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {a \left (x^2+2\right )}{a x^4+2 a x^2+a+b}dx}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \int \frac {x^2+2}{a x^4+2 a x^2+a+b}dx}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 1483

\(\displaystyle -\frac {a \left (\frac {\int \frac {2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x+2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \left (\frac {\int \frac {2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x+2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 1142

\(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}-\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {a \left (\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {a}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {a}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {a \left (\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\)

Input:

Int[1/(x^2*(a + b + 2*a*x^2 + a*x^4)),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1443

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim 
p[1/(a*d^2*(m + 1))   Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* 
x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 
- 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.18 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.49

method	result	size
risch	\(-\frac {1}{\left (a +b \right ) x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b^{2}+3 a^{2} b^{3}+3 b^{4} a +b^{5}\right ) \textit {\_Z}^{4}+\left (-2 a^{2} b +6 b^{2} a \right ) \textit {\_Z}^{2}+a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-b \,a^{4}+2 a^{3} b^{2}+12 a^{2} b^{3}+14 b^{4} a +5 b^{5}\right ) \textit {\_R}^{4}+\left (a^{3}-6 a^{2} b +25 b^{2} a \right ) \textit {\_R}^{2}+4 a \right ) x +\left (-3 a^{3} b -5 a^{2} b^{2}-b^{3} a +b^{4}\right ) \textit {\_R}^{3}\right )\right )}{4}\)	\(162\)
default	\(\text {Expression too large to display}\)	\(1262\)

Input:

int(1/x^2/(a*x^4+2*a*x^2+a+b),x,method=_RETURNVERBOSE)
 

Output:

-1/(a+b)/x+1/4*sum(_R*ln(((-a^4*b+2*a^3*b^2+12*a^2*b^3+14*a*b^4+5*b^5)*_R^ 
4+(a^3-6*a^2*b+25*a*b^2)*_R^2+4*a)*x+(-3*a^3*b-5*a^2*b^2-a*b^3+b^4)*_R^3), 
_R=RootOf((a^3*b^2+3*a^2*b^3+3*a*b^4+b^5)*_Z^4+(-2*a^2*b+6*a*b^2)*_Z^2+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1582 vs. \(2 (228) = 456\).

Time = 0.10 (sec) , antiderivative size = 1582, normalized size of antiderivative = 4.82 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(a*x^4+2*a*x^2+a+b),x, algorithm="fricas")
 

Output:

1/4*((a + b)*x*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqr 
t(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 
+ 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))*log(- 
(3*a^2 - a*b)*x + (6*a^2*b - 2*a*b^2 + (a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5) 
*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3* 
b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 
 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15* 
a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 
3*a*b^3 + b^4))) - (a + b)*x*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 + 3*a* 
b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 
 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 
 + b^4))*log(-(3*a^2 - a*b)*x - (6*a^2*b - 2*a*b^2 + (a^4*b + 2*a^3*b^2 - 
2*a*b^4 - b^5)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4 
*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b + (a^3 
*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6 
*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b 
+ 3*a^2*b^2 + 3*a*b^3 + b^4))) + (a + b)*x*sqrt((a^2 - 3*a*b - (a^3*b + 3* 
a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^ 
2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2 
*b^2 + 3*a*b^3 + b^4))*log(-(3*a^2 - a*b)*x + (6*a^2*b - 2*a*b^2 - (a^4...
 

Sympy [A] (verification not implemented)

Time = 1.86 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} b^{2} + 768 a^{2} b^{3} + 768 a b^{4} + 256 b^{5}\right ) + t^{2} \left (- 32 a^{2} b + 96 a b^{2}\right ) + a, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{4} b - 128 t^{3} a^{3} b^{2} + 128 t^{3} a b^{4} + 64 t^{3} b^{5} + 4 t a^{3} - 40 t a^{2} b + 20 t a b^{2}}{3 a^{2} - a b} \right )} \right )\right )} - \frac {1}{x \left (a + b\right )} \] Input:

integrate(1/x**2/(a*x**4+2*a*x**2+a+b),x)
 

Output:

RootSum(_t**4*(256*a**3*b**2 + 768*a**2*b**3 + 768*a*b**4 + 256*b**5) + _t 
**2*(-32*a**2*b + 96*a*b**2) + a, Lambda(_t, _t*log(x + (-64*_t**3*a**4*b 
- 128*_t**3*a**3*b**2 + 128*_t**3*a*b**4 + 64*_t**3*b**5 + 4*_t*a**3 - 40* 
_t*a**2*b + 20*_t*a*b**2)/(3*a**2 - a*b)))) - 1/(x*(a + b))
 

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + 2 \, a x^{2} + a + b\right )} x^{2}} \,d x } \] Input:

integrate(1/x^2/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")
 

Output:

-a*integrate((x^2 + 2)/(a*x^4 + 2*a*x^2 + a + b), x)/(a + b) - 1/((a + b)* 
x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (228) = 456\).

Time = 0.18 (sec) , antiderivative size = 742, normalized size of antiderivative = 2.26 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx =\text {Too large to display} \] Input:

integrate(1/x^2/(a*x^4+2*a*x^2+a+b),x, algorithm="giac")
 

Output:

1/2*((3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a*b + 4*sqrt(a^2 - sqrt(-a*b)* 
a)*sqrt(-a*b)*b^2)*(a + b)^2*abs(a) - 2*(3*sqrt(a^2 - sqrt(-a*b)*a)*a^3*b 
+ 7*sqrt(a^2 - sqrt(-a*b)*a)*a^2*b^2 + 4*sqrt(a^2 - sqrt(-a*b)*a)*a*b^3)*a 
bs(a)*abs(-a - b) - (3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^4 + 10*sqrt(a 
^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^3*b + 11*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a* 
b)*a^2*b^2 + 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a*b^3)*abs(a))*arctan(2 
*sqrt(1/2)*x/sqrt((2*a^2 + 2*a*b + sqrt(-4*(a^2 + 2*a*b + b^2)*(a^2 + a*b) 
 + 4*(a^2 + a*b)^2))/(a^2 + a*b)))/((3*a^6*b + 13*a^5*b^2 + 21*a^4*b^3 + 1 
5*a^3*b^4 + 4*a^2*b^5)*abs(-a - b)) - 1/2*((3*sqrt(a^2 + sqrt(-a*b)*a)*sqr 
t(-a*b)*a*b + 4*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*b^2)*(a + b)^2*abs(a) 
+ 2*(3*sqrt(a^2 + sqrt(-a*b)*a)*a^3*b + 7*sqrt(a^2 + sqrt(-a*b)*a)*a^2*b^2 
 + 4*sqrt(a^2 + sqrt(-a*b)*a)*a*b^3)*abs(a)*abs(-a - b) - (3*sqrt(a^2 + sq 
rt(-a*b)*a)*sqrt(-a*b)*a^4 + 10*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^3*b 
+ 11*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^2*b^2 + 4*sqrt(a^2 + sqrt(-a*b) 
*a)*sqrt(-a*b)*a*b^3)*abs(a))*arctan(2*sqrt(1/2)*x/sqrt((2*a^2 + 2*a*b - s 
qrt(-4*(a^2 + 2*a*b + b^2)*(a^2 + a*b) + 4*(a^2 + a*b)^2))/(a^2 + a*b)))/( 
(3*a^6*b + 13*a^5*b^2 + 21*a^4*b^3 + 15*a^3*b^4 + 4*a^2*b^5)*abs(-a - b)) 
- 1/((a + b)*x)
 

Mupad [B] (verification not implemented)

Time = 18.54 (sec) , antiderivative size = 2848, normalized size of antiderivative = 8.68 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\text {Too large to display} \] Input:

int(1/(x^2*(a + b + 2*a*x^2 + a*x^4)),x)
 

Output:

- 1/(x*(a + b)) - atan((((-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b 
^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 3 
2*a^4*b^5 + 128*a^5*b^4 + 192*a^6*b^3 + 128*a^7*b^2 + x*(-(3*a*b^2 - a^2*b 
 - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + 
 a^3*b^2)))^(1/2)*(64*a^9*b + 64*a^4*b^6 + 320*a^5*b^5 + 640*a^6*b^4 + 640 
*a^7*b^3 + 320*a^8*b^2)) - x*(8*a^7*b + 4*a^8 - 4*a^4*b^4 - 8*a^5*b^3))*(- 
(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b 
^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i - ((-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^ 
(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2 
)*(32*a^8*b + 32*a^4*b^5 + 128*a^5*b^4 + 192*a^6*b^3 + 128*a^7*b^2 - x*(-( 
3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^ 
5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b + 64*a^4*b^6 + 320*a^5*b^5 + 64 
0*a^6*b^4 + 640*a^7*b^3 + 320*a^8*b^2)) + x*(8*a^7*b + 4*a^8 - 4*a^4*b^4 - 
 8*a^5*b^3))*(-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/( 
16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i)/(6*a^6*b + 2*a^7 + (( 
-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + 
b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 32*a^4*b^5 + 128*a^5*b^4 + 
192*a^6*b^3 + 128*a^7*b^2 + x*(-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b* 
(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9* 
b + 64*a^4*b^6 + 320*a^5*b^5 + 640*a^6*b^4 + 640*a^7*b^3 + 320*a^8*b^2)...
 

Reduce [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.70 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx =\text {Too large to display} \] Input:

int(1/x^2/(a*x^4+2*a*x^2+a+b),x)
 

Output:

( - 2*sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*atan((sqrt(sqrt(a) 
*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sq 
rt(2)))*a*x + 2*sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*atan((sq 
rt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + 
b) + a)*sqrt(2)))*b*x + 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*at 
an((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqr 
t(a + b) + a)*sqrt(2)))*a*x + 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt 
(2)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt( 
a)*sqrt(a + b) + a)*sqrt(2)))*b*x + 2*sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) 
 + a)*sqrt(2)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/( 
sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a*x - 2*sqrt(a + b)*sqrt(sqrt(a)*s 
qrt(a + b) + a)*sqrt(2)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sq 
rt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*b*x - 2*sqrt(a)*sqrt(sqr 
t(a)*sqrt(a + b) + a)*sqrt(2)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) 
+ 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a*x - 2*sqrt(a)*sq 
rt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sq 
rt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*b*x + sqrt(a 
 + b)*sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*log( - sqrt(sqrt(a)*sqrt(a + b 
) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a*x - sqrt(a + b)*sqrt(sqrt 
(a)*sqrt(a + b) - a)*sqrt(2)*log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(...