Integrand size = 27, antiderivative size = 96 \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{\sqrt {2}} \]
-1/2*arctan(2^(1/2)*x*(x^6+x^2)^(1/4)/(-x^2+(x^6+x^2)^(1/2)))*2^(1/2)-1/2* arctanh((1/2*2^(1/2)*x^2+1/2*(x^6+x^2)^(1/2)*2^(1/2))/x/(x^6+x^2)^(1/4))*2 ^(1/2)
Time = 0.00 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.07 \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {\sqrt {x} \sqrt [4]{1+x^4} \left (\arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{-x+\sqrt {1+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x+\sqrt {1+x^4}}\right )\right )}{\sqrt {2} \sqrt [4]{x^2+x^6}} \]
-((Sqrt[x]*(1 + x^4)^(1/4)*(ArcTan[(Sqrt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(-x + Sqrt[1 + x^4])] + ArcTanh[(Sqrt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(x + Sqrt[1 + x^4])]))/(Sqrt[2]*(x^2 + x^6)^(1/4)))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.84 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.71, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2467, 25, 2035, 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.
| \(\displaystyle \int \frac {x^4-1}{\left (x^4+x^2+1\right ) \sqrt [4]{x^6+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4+1} \int -\frac {1-x^4}{\sqrt {x} \sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}dx}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {1-x^4}{\sqrt {x} \sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}dx}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \frac {1-x^4}{\sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \left (\frac {x^2+2}{\sqrt [4]{x^4+1} \left (x^4+x^2+1\right )}-\frac {1}{\sqrt [4]{x^4+1}}\right )d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \left (\sqrt {x} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )+\sqrt {x} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )+\frac {\left (-\sqrt {3}+i\right ) x^{5/2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{5 \left (\sqrt {3}+i\right )}+\frac {\left (\sqrt {3}+i\right ) x^{5/2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{5 \left (-\sqrt {3}+i\right )}-\sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},-x^4\right )\right )}{\sqrt [4]{x^6+x^2}}\) |
(-2*Sqrt[x]*(1 + x^4)^(1/4)*(Sqrt[x]*AppellF1[1/8, 1, 1/4, 9/8, (-2*x^4)/( 1 - I*Sqrt[3]), -x^4] + Sqrt[x]*AppellF1[1/8, 1, 1/4, 9/8, (-2*x^4)/(1 + I *Sqrt[3]), -x^4] + ((I - Sqrt[3])*x^(5/2)*AppellF1[5/8, 1/4, 1, 13/8, -x^4 , (-2*x^4)/(1 - I*Sqrt[3])])/(5*(I + Sqrt[3])) + ((I + Sqrt[3])*x^(5/2)*Ap pellF1[5/8, 1/4, 1, 13/8, -x^4, (-2*x^4)/(1 + I*Sqrt[3])])/(5*(I - Sqrt[3] )) - Sqrt[x]*Hypergeometric2F1[1/8, 1/4, 9/8, -x^4]))/(x^2 + x^6)^(1/4)
3.14.35.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
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]
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]
Time = 0.00 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right )}{4}\) | \(124\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x -2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right )}{2}\) | \(237\) |
1/4*2^(1/2)*(ln((-(x^2*(x^4+1))^(1/4)*2^(1/2)*x+x^2+(x^2*(x^4+1))^(1/2))/( (x^2*(x^4+1))^(1/4)*2^(1/2)*x+x^2+(x^2*(x^4+1))^(1/2)))+2*arctan(((x^2*(x^ 4+1))^(1/4)*2^(1/2)+x)/x)+2*arctan(((x^2*(x^4+1))^(1/4)*2^(1/2)-x)/x))
Result contains complex when optimal does not.
Time = 34.64 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.18 \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{5} - \left (i + 1\right ) \, x^{3} + \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{5} + \left (i + 1\right ) \, x^{3} - \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{5} + \left (i - 1\right ) \, x^{3} - \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{5} - \left (i - 1\right ) \, x^{3} + \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) \]
(1/8*I + 1/8)*sqrt(2)*log((4*I*(x^6 + x^2)^(1/4)*x^2 - (2*I - 2)*sqrt(2)*s qrt(x^6 + x^2)*x + sqrt(2)*((I + 1)*x^5 - (I + 1)*x^3 + (I + 1)*x) - 4*(x^ 6 + x^2)^(3/4))/(x^5 + x^3 + x)) - (1/8*I + 1/8)*sqrt(2)*log((4*I*(x^6 + x ^2)^(1/4)*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^6 + x^2)*x + sqrt(2)*(-(I + 1)*x^ 5 + (I + 1)*x^3 - (I + 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) - (1/ 8*I - 1/8)*sqrt(2)*log((-4*I*(x^6 + x^2)^(1/4)*x^2 + (2*I + 2)*sqrt(2)*sqr t(x^6 + x^2)*x + sqrt(2)*(-(I - 1)*x^5 + (I - 1)*x^3 - (I - 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) + (1/8*I - 1/8)*sqrt(2)*log((-4*I*(x^6 + x ^2)^(1/4)*x^2 - (2*I + 2)*sqrt(2)*sqrt(x^6 + x^2)*x + sqrt(2)*((I - 1)*x^5 - (I - 1)*x^3 + (I - 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x))
\[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral((x - 1)*(x + 1)*(x**2 + 1)/((x**2*(x**4 + 1))**(1/4)*(x**2 - x + 1)*(x**2 + x + 1)), x)
\[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}} \,d x } \]
\[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int \frac {x^4-1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^4+x^2+1\right )} \,d x \]