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

Optimal result

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

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 477, normalized size of antiderivative = 7.69, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2081, 1283, 1442, 399, 246, 218, 212, 209, 385, 214, 211} \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=-\frac {\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \left (\sqrt {3}+i\right ) \sqrt {x} \sqrt [4]{x^2+1} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\left (-\sqrt {3}+i\right ) \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {x} \sqrt [4]{x^2+1} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \left (\sqrt {3}+i\right ) \sqrt {x} \sqrt [4]{x^2+1} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\left (-\sqrt {3}+i\right ) \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {x} \sqrt [4]{x^2+1} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}} \]

[In]

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

[Out]

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

Rule 209

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

Rule 211

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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 218

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[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 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(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 1283

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With
[{k = Denominator[m]}, Dist[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] && Fra
ctionQ[m] && IntegerQ[p]

Rule 1442

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

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {\left (1+x^2\right )^{3/4}}{\sqrt {x} \left (1-x^2+x^4\right )} \, dx}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{1-x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{-1-i \sqrt {3}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}+\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{-1+i \sqrt {3}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (2 i \left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}-\frac {\left (2 i \left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-1-i \sqrt {3}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (2 i \left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}-\frac {\left (2 i \left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-3-i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}+\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i+\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i+\sqrt {3}\right )} \sqrt [4]{x^2+x^4}} \\ & = -\frac {\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^4}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

-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))

Maple [N/A] (verified)

Time = 35.84 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{4}-3}\right )}{2}\)	\(51\)
trager	\(\text {Expression too large to display}\)	\(1627\)

[In]

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

[Out]

-1/2*sum(_R^3*ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(2*_R^4-3),_R=RootOf(_Z^8-3*_Z^4+3))

Fricas [C] (verification not implemented)

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

Time = 2.86 (sec) , antiderivative size = 1566, normalized size of antiderivative = 25.26 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(1/24*I + 1/24)*3^(7/8)*sqrt(2)*(-1)^(1/8)*log((3^(7/8)*sqrt(2)*(-1)^(1/8)*((35*I + 35)*x^5 - (17*I + 17)*x^3
- (13*I + 13)*x) - 3*3^(3/8)*sqrt(2)*(-1)^(5/8)*(-(7*I + 7)*x^5 - (19*I + 19)*x^3 - (3*I + 3)*x) - 12*(x^4 + x
^2)^(3/4)*(2*x^2 + sqrt(3)*(-8*I*x^2 + 5*I) + 11) - 6*sqrt(x^4 + x^2)*(3^(5/8)*sqrt(2)*(-1)^(3/8)*((3*I - 3)*x
^3 - (8*I - 8)*x) + 3^(1/8)*sqrt(2)*(-1)^(7/8)*((13*I - 13)*x^3 - (2*I - 2)*x)) - 12*(x^4 + x^2)^(1/4)*(3^(1/4
)*(-1)^(3/4)*(11*I*x^4 - 13*I*x^2) + 3^(3/4)*(-1)^(1/4)*(-5*I*x^4 - 3*I*x^2)))/(x^5 - x^3 + x)) - (1/24*I - 1/
24)*3^(7/8)*sqrt(2)*(-1)^(1/8)*log((3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(35*I - 35)*x^5 + (17*I - 17)*x^3 + (13*I - 1
3)*x) - 3*3^(3/8)*sqrt(2)*(-1)^(5/8)*((7*I - 7)*x^5 + (19*I - 19)*x^3 + (3*I - 3)*x) - 12*(x^4 + x^2)^(3/4)*(2
*x^2 + sqrt(3)*(-8*I*x^2 + 5*I) + 11) - 6*sqrt(x^4 + x^2)*(3^(5/8)*sqrt(2)*(-1)^(3/8)*(-(3*I + 3)*x^3 + (8*I +
8)*x) + 3^(1/8)*sqrt(2)*(-1)^(7/8)*(-(13*I + 13)*x^3 + (2*I + 2)*x)) - 12*(x^4 + x^2)^(1/4)*(3^(3/4)*(-1)^(1/
4)*(5*I*x^4 + 3*I*x^2) + 3^(1/4)*(-1)^(3/4)*(-11*I*x^4 + 13*I*x^2)))/(x^5 - x^3 + x)) - (1/24*I + 1/24)*3^(7/8
)*sqrt(2)*(-1)^(1/8)*log(-(3*3^(3/8)*sqrt(2)*(-1)^(5/8)*((7*I + 7)*x^5 + (19*I + 19)*x^3 + (3*I + 3)*x) - 3^(7
/8)*sqrt(2)*(-1)^(1/8)*(-(35*I + 35)*x^5 + (17*I + 17)*x^3 + (13*I + 13)*x) + 12*(x^4 + x^2)^(3/4)*(2*x^2 + sq
rt(3)*(-8*I*x^2 + 5*I) + 11) + 6*sqrt(x^4 + x^2)*(3^(1/8)*sqrt(2)*(-1)^(7/8)*(-(13*I - 13)*x^3 + (2*I - 2)*x)
+ 3^(5/8)*sqrt(2)*(-1)^(3/8)*(-(3*I - 3)*x^3 + (8*I - 8)*x)) + 12*(x^4 + x^2)^(1/4)*(3^(1/4)*(-1)^(3/4)*(11*I*
x^4 - 13*I*x^2) + 3^(3/4)*(-1)^(1/4)*(-5*I*x^4 - 3*I*x^2)))/(x^5 - x^3 + x)) + (1/24*I - 1/24)*3^(7/8)*sqrt(2)
*(-1)^(1/8)*log(-(3*3^(3/8)*sqrt(2)*(-1)^(5/8)*(-(7*I - 7)*x^5 - (19*I - 19)*x^3 - (3*I - 3)*x) - 3^(7/8)*sqrt
(2)*(-1)^(1/8)*((35*I - 35)*x^5 - (17*I - 17)*x^3 - (13*I - 13)*x) + 12*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt(3)*(-8
*I*x^2 + 5*I) + 11) + 6*sqrt(x^4 + x^2)*(3^(1/8)*sqrt(2)*(-1)^(7/8)*((13*I + 13)*x^3 - (2*I + 2)*x) + 3^(5/8)*
sqrt(2)*(-1)^(3/8)*((3*I + 3)*x^3 - (8*I + 8)*x)) + 12*(x^4 + x^2)^(1/4)*(3^(3/4)*(-1)^(1/4)*(5*I*x^4 + 3*I*x^
2) + 3^(1/4)*(-1)^(3/4)*(-11*I*x^4 + 13*I*x^2)))/(x^5 - x^3 + x)) - 1/12*3^(7/8)*(-1)^(1/8)*log(-(3^(7/8)*(-1)
^(1/8)*(35*x^5 - 17*x^3 - 13*x) - 3*3^(3/8)*(-1)^(5/8)*(7*x^5 + 19*x^3 + 3*x) + 6*(x^4 + x^2)^(3/4)*(2*x^2 + s
qrt(3)*(8*I*x^2 - 5*I) + 11) + 6*sqrt(x^4 + x^2)*(3^(1/8)*(-1)^(7/8)*(13*x^3 - 2*x) - 3^(5/8)*(-1)^(3/8)*(3*x^
3 - 8*x)) - 6*(x^4 + x^2)^(1/4)*(3^(1/4)*(-1)^(3/4)*(11*x^4 - 13*x^2) + 3^(3/4)*(-1)^(1/4)*(5*x^4 + 3*x^2)))/(
x^5 - x^3 + x)) + 1/12*3^(7/8)*(-1)^(1/8)*log((3^(7/8)*(-1)^(1/8)*(35*x^5 - 17*x^3 - 13*x) - 3*3^(3/8)*(-1)^(5
/8)*(7*x^5 + 19*x^3 + 3*x) - 6*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt(3)*(8*I*x^2 - 5*I) + 11) + 6*sqrt(x^4 + x^2)*(3
^(1/8)*(-1)^(7/8)*(13*x^3 - 2*x) - 3^(5/8)*(-1)^(3/8)*(3*x^3 - 8*x)) + 6*(x^4 + x^2)^(1/4)*(3^(1/4)*(-1)^(3/4)
*(11*x^4 - 13*x^2) + 3^(3/4)*(-1)^(1/4)*(5*x^4 + 3*x^2)))/(x^5 - x^3 + x)) + 1/12*I*3^(7/8)*(-1)^(1/8)*log((3^
(7/8)*(-1)^(1/8)*(35*I*x^5 - 17*I*x^3 - 13*I*x) - 3*3^(3/8)*(-1)^(5/8)*(7*I*x^5 + 19*I*x^3 + 3*I*x) - 6*(x^4 +
x^2)^(3/4)*(2*x^2 + sqrt(3)*(8*I*x^2 - 5*I) + 11) - 6*sqrt(x^4 + x^2)*(3^(1/8)*(-1)^(7/8)*(13*I*x^3 - 2*I*x)
+ 3^(5/8)*(-1)^(3/8)*(-3*I*x^3 + 8*I*x)) - 6*(x^4 + x^2)^(1/4)*(3^(1/4)*(-1)^(3/4)*(11*x^4 - 13*x^2) + 3^(3/4)
*(-1)^(1/4)*(5*x^4 + 3*x^2)))/(x^5 - x^3 + x)) - 1/12*I*3^(7/8)*(-1)^(1/8)*log(-(3*3^(3/8)*(-1)^(5/8)*(-7*I*x^
5 - 19*I*x^3 - 3*I*x) - 3^(7/8)*(-1)^(1/8)*(-35*I*x^5 + 17*I*x^3 + 13*I*x) + 6*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt
(3)*(8*I*x^2 - 5*I) + 11) + 6*sqrt(x^4 + x^2)*(3^(5/8)*(-1)^(3/8)*(3*I*x^3 - 8*I*x) + 3^(1/8)*(-1)^(7/8)*(-13*
I*x^3 + 2*I*x)) + 6*(x^4 + x^2)^(1/4)*(3^(1/4)*(-1)^(3/4)*(11*x^4 - 13*x^2) + 3^(3/4)*(-1)^(1/4)*(5*x^4 + 3*x^
2)))/(x^5 - x^3 + x))

Sympy [N/A]

Not integrable

Time = 2.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.42 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{2} + 1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - x^{2} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.74 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}} \,d x } \]

[In]

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

[Out]

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 = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^2+1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-x^2+1\right )} \,d x \]

[In]

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

[Out]

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