Integrand size = 27, antiderivative size = 118 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\frac {2 \sqrt [4]{x^2+x^6}}{x}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{\sqrt [4]{2}} \]
2*(x^6+x^2)^(1/4)/x-1/2*arctan(2^(3/4)*x*(x^6+x^2)^(1/4)/(2^(1/2)*x^2-(x^6 +x^2)^(1/2)))*2^(3/4)-1/2*arctanh((1/2*x^2*2^(3/4)+1/2*(x^6+x^2)^(1/2)*2^( 1/4))/x/(x^6+x^2)^(1/4))*2^(3/4)
Time = 0.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=-\frac {\sqrt [4]{x^2+x^6} \left (-4 \sqrt [4]{1+x^4}+2^{3/4} \sqrt {x} \arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )+2^{3/4} \sqrt {x} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{2 x \sqrt [4]{1+x^4}} \]
-1/2*((x^2 + x^6)^(1/4)*(-4*(1 + x^4)^(1/4) + 2^(3/4)*Sqrt[x]*ArcTan[(2^(3 /4)*Sqrt[x]*(1 + x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^4])] + 2^(3/4)*Sqrt[x ]*ArcTanh[(2*2^(1/4)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x + Sqrt[2]*Sqrt[1 + x^4] )]))/(x*(1 + x^4)^(1/4))
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.
| \(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt [4]{x^6+x^2}}{x^2 \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^6+x^2} \int -\frac {\left (1-x^2\right ) \sqrt [4]{x^4+1}}{x^{3/2} \left (x^2+1\right )}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^6+x^2} \int \frac {\left (1-x^2\right ) \sqrt [4]{x^4+1}}{x^{3/2} \left (x^2+1\right )}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {\left (1-x^2\right ) \sqrt [4]{x^4+1}}{x \left (x^2+1\right )}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1}}{x}-\frac {2 x \sqrt [4]{x^4+1}}{x^2+1}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \left (-2 \int \frac {x \sqrt [4]{x^4+1}}{x^2+1}d\sqrt {x}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-\frac {1}{8},\frac {7}{8},-x^4\right )}{\sqrt {x}}\right )}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
3.18.55.3.1 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_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.00 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.35
| method | result | size |
| pseudoelliptic | \(\frac {-\ln \left (\frac {2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{-2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) 2^{\frac {3}{4}} x -2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {3}{4}} x -2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {3}{4}} x +8 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{4 x}\) | \(159\) |
1/4*(-ln((2^(3/4)*(x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^2+(x^2*(x^4+1))^(1/2))/( -2^(3/4)*(x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^2+(x^2*(x^4+1))^(1/2)))*2^(3/4)*x -2*arctan((2^(1/4)*(x^2*(x^4+1))^(1/4)+x)/x)*2^(3/4)*x-2*arctan((2^(1/4)*( x^2*(x^4+1))^(1/4)-x)/x)*2^(3/4)*x+8*(x^2*(x^4+1))^(1/4))/x
Result contains complex when optimal does not.
Time = 4.64 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.81 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\frac {\left (-2\right )^{\frac {1}{4}} x \log \left (\frac {4 \, \left (-2\right )^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2 \, \left (-2\right )^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {-2} {\left (x^{5} - 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} + 2 \, x^{3} + x}\right ) - \left (-2\right )^{\frac {1}{4}} x \log \left (-\frac {4 \, \left (-2\right )^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2 \, \left (-2\right )^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {-2} {\left (x^{5} - 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} + 2 \, x^{3} + x}\right ) + i \, \left (-2\right )^{\frac {1}{4}} x \log \left (\frac {4 i \, \left (-2\right )^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 i \, \left (-2\right )^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {-2} {\left (x^{5} - 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} + 2 \, x^{3} + x}\right ) - i \, \left (-2\right )^{\frac {1}{4}} x \log \left (\frac {-4 i \, \left (-2\right )^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2 i \, \left (-2\right )^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {-2} {\left (x^{5} - 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} + 2 \, x^{3} + x}\right ) + 8 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}}{4 \, x} \]
1/4*((-2)^(1/4)*x*log((4*(-2)^(1/4)*(x^6 + x^2)^(1/4)*x^2 - 2*(-2)^(3/4)*( x^6 + x^2)^(3/4) + sqrt(-2)*(x^5 - 2*x^3 + x) - 4*sqrt(x^6 + x^2)*x)/(x^5 + 2*x^3 + x)) - (-2)^(1/4)*x*log(-(4*(-2)^(1/4)*(x^6 + x^2)^(1/4)*x^2 - 2* (-2)^(3/4)*(x^6 + x^2)^(3/4) - sqrt(-2)*(x^5 - 2*x^3 + x) + 4*sqrt(x^6 + x ^2)*x)/(x^5 + 2*x^3 + x)) + I*(-2)^(1/4)*x*log((4*I*(-2)^(1/4)*(x^6 + x^2) ^(1/4)*x^2 + 2*I*(-2)^(3/4)*(x^6 + x^2)^(3/4) - sqrt(-2)*(x^5 - 2*x^3 + x) - 4*sqrt(x^6 + x^2)*x)/(x^5 + 2*x^3 + x)) - I*(-2)^(1/4)*x*log((-4*I*(-2) ^(1/4)*(x^6 + x^2)^(1/4)*x^2 - 2*I*(-2)^(3/4)*(x^6 + x^2)^(3/4) - sqrt(-2) *(x^5 - 2*x^3 + x) - 4*sqrt(x^6 + x^2)*x)/(x^5 + 2*x^3 + x)) + 8*(x^6 + x^ 2)^(1/4))/x
\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{2} \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^2-1\right )}{x^2\,\left (x^2+1\right )} \,d x \]