\(\int \frac {1+x^2}{(1-x^2+x^4) \sqrt [4]{x^2+x^4}} \, dx\) [815]

Optimal result

Integrand size = 29, antiderivative size = 62 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=-\frac {1}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3+2 \text {$\#$1}^4}\&\right ] \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.50 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=-\frac {\left (x^2+x^4\right )^{3/4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}^3+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^3}{-3+2 \text {$\#$1}^4}\&\right ]}{2 x^{3/2} \left (1+x^2\right )^{3/4}} \] Input:

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

Output:

-1/2*((x^2 + x^4)^(3/4)*RootSum[3 - 3*#1^4 + #1^8 & , (-(Log[Sqrt[x]]*#1^3 
) + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1]*#1^3)/(-3 + 2*#1^4) & ])/(x^(3/2)*(1 
 + x^2)^(3/4))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.71 (sec) , antiderivative size = 522, normalized size of antiderivative = 8.42, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2467, 1592, 1758, 25, 916, 770, 756, 216, 219, 902, 756, 218, 221}

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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4}}{\sqrt {x} \left (x^4-x^2+1\right )}dx}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 1592

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4}}{x^4-x^2+1}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 1758

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \int -\frac {\left (x^2+1\right )^{3/4}}{-2 x^2-i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}-\frac {2 i \int -\frac {\left (x^2+1\right )^{3/4}}{-2 x^2+i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \int \frac {\left (x^2+1\right )^{3/4}}{-2 x^2+i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}-\frac {2 i \int \frac {\left (x^2+1\right )^{3/4}}{-2 x^2-i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 916

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (-\frac {1}{2} \int \frac {1}{\sqrt [4]{x^2+1}}d\sqrt {x}+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{2} \int \frac {1}{\sqrt [4]{x^2+1}}d\sqrt {x}+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (-\frac {1}{2} \int \frac {1}{1-x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{2} \int \frac {1}{1-x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \int \frac {1}{x+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \int \frac {1}{x+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 902

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{-\left (\left (3+i \sqrt {3}\right ) x^2\right )+i \sqrt {3}+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{-\left (\left (3-i \sqrt {3}\right ) x^2\right )-i \sqrt {3}+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {-\sqrt {3}+i}}+\frac {i \int \frac {1}{\sqrt {3 i-\sqrt {3}} x+\sqrt {i-\sqrt {3}}}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {-\sqrt {3}+i}}\right )\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {\sqrt {3}+i}}+\frac {i \int \frac {1}{\sqrt {3 i+\sqrt {3}} x+\sqrt {i+\sqrt {3}}}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {\sqrt {3}+i}}\right )\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {-\sqrt {3}+i}}+\frac {i \sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (-\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {\sqrt {3}+i}}+\frac {i \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \left (\frac {i \sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (-\sqrt {3}+i\right )}+\frac {i \sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (-\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \left (\frac {i \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (\sqrt {3}+i\right )}+\frac {i \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\)

Input:

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

Output:

(2*Sqrt[x]*(1 + x^2)^(1/4)*(((2*I)*((-1/2*ArcTan[Sqrt[x]/(1 + x^2)^(1/4)] 
- ArcTanh[Sqrt[x]/(1 + x^2)^(1/4)]/2)/2 + ((3 + I*Sqrt[3])*(((I/2)*((I - S 
qrt[3])/(3*I - Sqrt[3]))^(1/4)*ArcTan[Sqrt[x]/(((I - Sqrt[3])/(3*I - Sqrt[ 
3]))^(1/4)*(1 + x^2)^(1/4))])/(I - Sqrt[3]) + ((I/2)*((I - Sqrt[3])/(3*I - 
 Sqrt[3]))^(1/4)*ArcTanh[Sqrt[x]/(((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/4)*(1 
 + x^2)^(1/4))])/(I - Sqrt[3])))/2))/Sqrt[3] - ((2*I)*((-1/2*ArcTan[Sqrt[x 
]/(1 + x^2)^(1/4)] - ArcTanh[Sqrt[x]/(1 + x^2)^(1/4)]/2)/2 + ((3 - I*Sqrt[ 
3])*(((I/2)*((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*ArcTan[Sqrt[x]/(((I + Sq 
rt[3])/(3*I + Sqrt[3]))^(1/4)*(1 + x^2)^(1/4))])/(I + Sqrt[3]) + ((I/2)*(( 
I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*ArcTanh[Sqrt[x]/(((I + Sqrt[3])/(3*I + 
 Sqrt[3]))^(1/4)*(1 + x^2)^(1/4))])/(I + Sqrt[3])))/2))/Sqrt[3]))/(x^2 + x 
^4)^(1/4)
 

Defintions of rubi rules used

rule 25

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

rule 216

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

rule 218

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

rule 219

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

rule 221

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

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 902

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su 
bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b 
, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
 

rule 916

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[b/d   Int[(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b* 
x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - 
a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]
 

rule 1592

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c 
_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f   Subst[ 
Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f^2))^q*(a + b*(x^(2*k)/f^k) + c*(x^( 
4*k)/f^4))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x 
] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
 

rule 1758

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r)   Int[(d + e*x 
^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r)   Int[(d + e*x^n)^q/(b + r + 
 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne 
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[q]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 36.55 (sec) , antiderivative size = 1627, normalized size of antiderivative = 26.24

\[\text {Expression too large to display}\]

Input:

int((x^2+1)/(x^4-x^2+1)/(x^4+x^2)^(1/4),x)
 

Output:

-1/6*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*ln((RootOf(_Z^2+RootOf(_Z^8+2187)^2) 
*RootOf(_Z^8+2187)^8*x^3+4*(x^4+x^2)^(1/2)*RootOf(_Z^8+2187)^6*RootOf(_Z^2 
+RootOf(_Z^8+2187)^2)*x-RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187) 
^8*x+264*(x^4+x^2)^(1/4)*RootOf(_Z^8+2187)^6*x^2+558*x^3*RootOf(_Z^2+RootO 
f(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4+756*(x^4+x^2)^(3/4)*RootOf(_Z^8+2187)^ 
4+4860*(x^4+x^2)^(1/2)*RootOf(_Z^8+2187)^2*RootOf(_Z^2+RootOf(_Z^8+2187)^2 
)*x+144*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4*x-7776*(x^4+x 
^2)^(1/4)*RootOf(_Z^8+2187)^2*x^2+23085*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x 
^3-67068*(x^4+x^2)^(3/4)+8505*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x)/(x^2*Roo 
tOf(_Z^8+2187)^4-RootOf(_Z^8+2187)^4-27*x^2-27)/x)+1/6*RootOf(_Z^8+2187)*l 
n((-RootOf(_Z^8+2187)^9*x^3+4*(x^4+x^2)^(1/2)*RootOf(_Z^8+2187)^7*x+x*Root 
Of(_Z^8+2187)^9-264*(x^4+x^2)^(1/4)*RootOf(_Z^8+2187)^6*x^2-558*x^3*RootOf 
(_Z^8+2187)^5+756*(x^4+x^2)^(3/4)*RootOf(_Z^8+2187)^4+4860*(x^4+x^2)^(1/2) 
*RootOf(_Z^8+2187)^3*x-144*RootOf(_Z^8+2187)^5*x+7776*(x^4+x^2)^(1/4)*Root 
Of(_Z^8+2187)^2*x^2-23085*RootOf(_Z^8+2187)*x^3-67068*(x^4+x^2)^(3/4)-8505 
*RootOf(_Z^8+2187)*x)/(x^2*RootOf(_Z^8+2187)^4-RootOf(_Z^8+2187)^4-27*x^2- 
27)/x)+1/6*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)) 
*ln(-(RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*Root 
Of(_Z^8+2187)^8*x^3+4*(x^4+x^2)^(1/2)*RootOf(_Z^8+2187)^5*RootOf(_Z^2-Root 
Of(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^2+RootOf(_Z^8...
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 2.77 (sec) , antiderivative size = 1398, normalized size of antiderivative = 22.55 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \] Input:

integrate((x^2+1)/(x^4-x^2+1)/(x^4+x^2)^(1/4),x, algorithm="fricas")
 

Output:

(1/8*I - 1/8)*sqrt(2)*(-1/3)^(1/8)*log(-(6*sqrt(2)*sqrt(x^4 + x^2)*((-1/3) 
^(7/8)*((13*I + 13)*x^3 - (2*I + 2)*x) + (-1/3)^(3/8)*((3*I + 3)*x^3 - (8* 
I + 8)*x)) + 4*(x^4 + x^2)^(3/4)*(2*x^2 - 3*sqrt(-1/3)*(8*x^2 - 5) + 11) + 
 sqrt(2)*(3*(-1/3)^(5/8)*(-(7*I - 7)*x^5 - (19*I - 19)*x^3 - (3*I - 3)*x) 
- (-1/3)^(1/8)*((35*I - 35)*x^5 - (17*I - 17)*x^3 - (13*I - 13)*x)) + 12*( 
x^4 + x^2)^(1/4)*((-1/3)^(3/4)*(-11*I*x^4 + 13*I*x^2) + (-1/3)^(1/4)*(5*I* 
x^4 + 3*I*x^2)))/(x^5 - x^3 + x)) - (1/8*I + 1/8)*sqrt(2)*(-1/3)^(1/8)*log 
(-(6*sqrt(2)*sqrt(x^4 + x^2)*((-1/3)^(7/8)*(-(13*I - 13)*x^3 + (2*I - 2)*x 
) + (-1/3)^(3/8)*(-(3*I - 3)*x^3 + (8*I - 8)*x)) + 4*(x^4 + x^2)^(3/4)*(2* 
x^2 - 3*sqrt(-1/3)*(8*x^2 - 5) + 11) + sqrt(2)*(3*(-1/3)^(5/8)*((7*I + 7)* 
x^5 + (19*I + 19)*x^3 + (3*I + 3)*x) - (-1/3)^(1/8)*(-(35*I + 35)*x^5 + (1 
7*I + 17)*x^3 + (13*I + 13)*x)) + 12*(x^4 + x^2)^(1/4)*((-1/3)^(3/4)*(11*I 
*x^4 - 13*I*x^2) + (-1/3)^(1/4)*(-5*I*x^4 - 3*I*x^2)))/(x^5 - x^3 + x)) + 
(1/8*I + 1/8)*sqrt(2)*(-1/3)^(1/8)*log(-(6*sqrt(2)*sqrt(x^4 + x^2)*((-1/3) 
^(7/8)*((13*I - 13)*x^3 - (2*I - 2)*x) + (-1/3)^(3/8)*((3*I - 3)*x^3 - (8* 
I - 8)*x)) + 4*(x^4 + x^2)^(3/4)*(2*x^2 - 3*sqrt(-1/3)*(8*x^2 - 5) + 11) + 
 sqrt(2)*(3*(-1/3)^(5/8)*(-(7*I + 7)*x^5 - (19*I + 19)*x^3 - (3*I + 3)*x) 
- (-1/3)^(1/8)*((35*I + 35)*x^5 - (17*I + 17)*x^3 - (13*I + 13)*x)) + 12*( 
x^4 + x^2)^(1/4)*((-1/3)^(3/4)*(11*I*x^4 - 13*I*x^2) + (-1/3)^(1/4)*(-5*I* 
x^4 - 3*I*x^2)))/(x^5 - x^3 + x)) - (1/8*I - 1/8)*sqrt(2)*(-1/3)^(1/8)*...
 

Sympy [N/A]

Not integrable

Time = 2.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.42 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{2} + 1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - x^{2} + 1\right )}\, dx \] Input:

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

Output:

Integral((x**2 + 1)/((x**2*(x**2 + 1))**(1/4)*(x**4 - x**2 + 1)), x)
 

Maxima [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.74 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}} \,d x } \] Input:

integrate((x^2+1)/(x^4-x^2+1)/(x^4+x^2)^(1/4),x, algorithm="maxima")
 

Output:

2/21*(8*x^5 - 7*(x^3 + x)*x^2 + 9*x^3 + x)/((x^(9/2) - x^(5/2) + sqrt(x))* 
(x^2 + 1)^(1/4)) + integrate(-4/21*(16*x^4 - 8*(x^4 + 2*x^2 + 1)*x^2 + 11* 
x^2 - 5)/((x^(17/2) - 2*x^(13/2) + 3*x^(9/2) - 2*x^(5/2) + sqrt(x))*(x^2 + 
 1)^(1/4)), x)
 

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}} \,d x } \] Input:

integrate((x^2+1)/(x^4-x^2+1)/(x^4+x^2)^(1/4),x, algorithm="giac")
 

Output:

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

Mupad [N/A]

Not integrable

Time = 7.71 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^2+1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-x^2+1\right )} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.13 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {-2 \sqrt {x}\, \left (x^{2}+1\right )^{\frac {1}{4}}+2 \sqrt {x^{2}+1}\, \left (\int \frac {\left (x^{2}+1\right )^{\frac {3}{4}}}{\sqrt {x}\, x^{8}+\sqrt {x}\, x^{6}+\sqrt {x}\, x^{2}+\sqrt {x}}d x \right )+2 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x^{5}}{x^{8}+x^{6}+x^{2}+1}d x \right )+3 \sqrt {x^{2}+1}\, \left (\int \frac {\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{4}} x}{x^{8}+x^{6}+x^{2}+1}d x \right )}{\sqrt {x^{2}+1}} \] Input:

int((x^2+1)/(x^4-x^2+1)/(x^4+x^2)^(1/4),x)
 

Output:

( - 2*sqrt(x)*(x**2 + 1)**(1/4) + 2*sqrt(x**2 + 1)*int((x**2 + 1)**(3/4)/( 
sqrt(x)*x**8 + sqrt(x)*x**6 + sqrt(x)*x**2 + sqrt(x)),x) + 2*sqrt(x**2 + 1 
)*int((sqrt(x)*(x**2 + 1)**(3/4)*x**5)/(x**8 + x**6 + x**2 + 1),x) + 3*sqr 
t(x**2 + 1)*int((sqrt(x)*(x**2 + 1)**(3/4)*x)/(x**8 + x**6 + x**2 + 1),x)) 
/sqrt(x**2 + 1)