Optimal. Leaf size=29 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^6-2 x^2-1}}\right )}{\sqrt {2}} \]
________________________________________________________________________________________
Rubi [F] time = 1.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx &=\int \left (\frac {2}{\sqrt {-1-2 x^2+x^6}}+\frac {3}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+3 \int \frac {1}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx\\ &=2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+3 \int \left (\frac {1}{3 \left (-1+x^2\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-2+x}{6 \left (1-x+x^2\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-2-x}{6 \left (1+x+x^2\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-2+x}{\left (1-x+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx\\ &=\frac {1}{2} \int \left (\frac {1+i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}+\frac {1-i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx+\frac {1}{2} \int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\int \left (\frac {1}{2 (-1+x) \sqrt {-1-2 x^2+x^6}}-\frac {1}{2 (1+x) \sqrt {-1-2 x^2+x^6}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{(-1+x) \sqrt {-1-2 x^2+x^6}} \, dx-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {-1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.06, size = 29, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1-2 x^2+x^6}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 36, normalized size = 1.24 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{6} - 2 \, x^{2} - 1} x}{x^{6} - 4 \, x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{6} + 1}{\sqrt {x^{6} - 2 \, x^{2} - 1} {\left (x^{6} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.30, size = 84, normalized size = 2.90
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}-4 \sqrt {x^{6}-2 x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{4}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{6} + 1}{\sqrt {x^{6} - 2 \, x^{2} - 1} {\left (x^{6} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x^6+1}{\left (x^6-1\right )\,\sqrt {x^6-2\,x^2-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x^{6} + 1}{\sqrt {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________