∫ −1+x2 ________________________________________(1+x2 )√ _________

Optimal. Leaf size=120 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4+14 \text {$\#$1}^2-16 \text {$\#$1}+5\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1} x-x^2+\sqrt {x^4+x^3-x^2-x+1}+1\right )+\text {$\#$1}^2 (-\log (x))-\log \left (-\text {$\#$1} x-x^2+\sqrt {x^4+x^3-x^2-x+1}+1\right )+\log (x)}{\text {$\#$1}^3+7 \text {$\#$1}-4}\& \right ] \]

________________________________________________________________________________________

Rubi [F] time = 0.41, 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+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx \end {gather*}

\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {1-x-x^2+x^3+x^4}}-\frac {2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ &=-\left (2 \int \left (\frac {i}{2 (i-x) \sqrt {1-x-x^2+x^3+x^4}}+\frac {i}{2 (i+x) \sqrt {1-x-x^2+x^3+x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ &=-\left (i \int \frac {1}{(i-x) \sqrt {1-x-x^2+x^3+x^4}} \, dx\right )-i \int \frac {1}{(i+x) \sqrt {1-x-x^2+x^3+x^4}} \, dx+\int \frac {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] time = 1.28, size = 3242, normalized size = 27.02 \begin {gather*} \text {Result too large to show} \end {gather*}

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 0.00, size = 120, normalized size = 1.00 \begin {gather*} \frac {1}{2} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

________________________________________________________________________________________


method	result	size

trager	\(-\frac {\RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {8112 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{5} x +3536 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3} x^{2}+4147 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3} x -3536 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3}-1820 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+85 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) x^{2}+95 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) x -85 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )-762 \sqrt {x^{4}+x^{3}-x^{2}-x +1}}{\left (13 x \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3 x -2\right )^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \ln \left (\frac {624 \RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{4} x -272 \RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} x^{2}+257 \RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} x +272 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right )+1820 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}-119 \RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x^{2}-7 \RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x +119 \RootOf \left (\textit {\_Z}^{2}+169 \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right )+78 \sqrt {x^{4}+x^{3}-x^{2}-x +1}}{\left (13 x \RootOf \left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3 x +2\right )^{2}}\right )}{26}\)	$574$
default	$\text {Expression too large to display}$	$92318$
elliptic	$\text {Expression too large to display}$	$105692$

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

________________________________________________________________________________________

mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+x^3-x^2-x+1}} \,d x \end {gather*}

________________________________________________________________________________________

sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}\, dx \end {gather*}

________________________________________________________________________________________

3.18.81 \(\int \frac {-1+x^2}{(1+x^2) \sqrt {1-x-x^2+x^3+x^4}} \, dx\)