\(\int \frac {-1+x^2}{(1+x^2) \sqrt {1-x-x^2+x^3+x^4}} \, dx\) [1780]

Optimal result

Integrand size = 33, antiderivative size = 120 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx \]

[In]

Int[(-1 + x^2)/((1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]

[Out]

Defer[Int][1/Sqrt[1 - x - x^2 + x^3 + x^4], x] - I*Defer[Int][1/((I - x)*Sqrt[1 - x - x^2 + x^3 + x^4]), x] -
I*Defer[Int][1/((I + x)*Sqrt[1 - x - x^2 + x^3 + x^4]), x]

Rubi steps \begin{align*} \text {integral}& = \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 [A] (verified)

Time = 0.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]

[In]

Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]

[Out]

RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*
#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/2

Maple [N/A] (verified)

Time = 8.11 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.19

method	result	size
default	\(\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {6+2 \sqrt {13}}\, \left (\sqrt {13}-\frac {7}{2}\right ) \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\arctan \left (\frac {35 \sqrt {13}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \sqrt {-2+\sqrt {13}}\, \left (-\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \sqrt {-2+\sqrt {13}}\right )}{\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\sqrt {13}+3\right ) \left (7 \sqrt {13}-26\right ) x}\)	\(263\)
pseudoelliptic	\(\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {6+2 \sqrt {13}}\, \left (\sqrt {13}-\frac {7}{2}\right ) \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\arctan \left (\frac {35 \sqrt {13}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \sqrt {-2+\sqrt {13}}\, \left (-\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \sqrt {-2+\sqrt {13}}\right )}{\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\sqrt {13}+3\right ) \left (7 \sqrt {13}-26\right ) x}\)	\(263\)
trager	\(-\frac {\operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-8112 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{5} x -3536 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3} x^{2}-4147 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3} x +1820 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3536 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{3}-85 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) x^{2}-95 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right ) x +762 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+85 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )}{{\left (13 x \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3 x -2\right )}^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \ln \left (\frac {-624 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{4} x +272 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} x^{2}-257 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} x +1820 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}-272 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right )+119 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x +78 \sqrt {x^{4}+x^{3}-x^{2}-x +1}-119 \operatorname {RootOf}\left (\textit {\_Z}^{2}+169 \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+78\right )}{{\left (13 x \operatorname {RootOf}\left (13 \textit {\_Z}^{4}+6 \textit {\_Z}^{2}+1\right )^{2}+3 x +2\right )}^{2}}\right )}{26}\)	\(575\)
elliptic	\(\text {Expression too large to display}\)	\(105692\)

[In]

int((x^2-1)/(x^2+1)/(x^4+x^3-x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*((x^4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)/((x^4+x^3-x^2-x+1)/x^2)^(1/2)*(13^(1/2)*x+x^2-3*x-1)*((6+
2*13^(1/2))^(1/2)*(13^(1/2)-7/2)*arctanh(4*((x^4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)/(6+2*13^(1/2))^(
1/2))+arctan(35/468*13^(1/2)*((x^4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)*(13^(1/2)+19/5)*(-2+13^(1/2))^
(1/2)*(-13^(1/2)*x+x^2-3*x-1)*(13^(1/2)-26/7)*(13^(1/2)*x+x^2-3*x-1)/(x^4+x^3-x^2-x+1))*(-2+13^(1/2))^(1/2))/(
13^(1/2)+3)/(7*13^(1/2)-26)/x

Fricas [C] (verification not implemented)

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

Time = 0.34 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.68 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{104} \, \sqrt {13} \sqrt {-8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {-8 i - 12} {\left (\left (544 i + 731\right ) \, x^{4} + \left (1524 i - 466\right ) \, x^{3} + \left (156 i - 1586\right ) \, x^{2} - \left (1524 i - 466\right ) \, x + 544 i + 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (140 i - 171\right ) \, x^{2} + \left (342 i + 280\right ) \, x + 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{104} \, \sqrt {13} \sqrt {8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {8 i - 12} {\left (-\left (544 i - 731\right ) \, x^{4} - \left (1524 i + 466\right ) \, x^{3} - \left (156 i + 1586\right ) \, x^{2} + \left (1524 i + 466\right ) \, x - 544 i + 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (140 i + 171\right ) \, x^{2} - \left (342 i - 280\right ) \, x - 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{104} \, \sqrt {13} \sqrt {8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {8 i - 12} {\left (\left (544 i - 731\right ) \, x^{4} + \left (1524 i + 466\right ) \, x^{3} + \left (156 i + 1586\right ) \, x^{2} - \left (1524 i + 466\right ) \, x + 544 i - 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (140 i + 171\right ) \, x^{2} - \left (342 i - 280\right ) \, x - 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{104} \, \sqrt {13} \sqrt {-8 i - 12} \log \left (\frac {\sqrt {13} \sqrt {-8 i - 12} {\left (-\left (544 i + 731\right ) \, x^{4} - \left (1524 i - 466\right ) \, x^{3} - \left (156 i - 1586\right ) \, x^{2} + \left (1524 i - 466\right ) \, x - 544 i - 731\right )} - 52 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (140 i - 171\right ) \, x^{2} + \left (342 i + 280\right ) \, x + 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) \]

[In]

integrate((x^2-1)/(x^2+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="fricas")

[Out]

1/104*sqrt(13)*sqrt(-8*I - 12)*log((sqrt(13)*sqrt(-8*I - 12)*((544*I + 731)*x^4 + (1524*I - 466)*x^3 + (156*I
- 1586)*x^2 - (1524*I - 466)*x + 544*I + 731) - 52*sqrt(x^4 + x^3 - x^2 - x + 1)*(-(140*I - 171)*x^2 + (342*I
+ 280)*x + 140*I - 171))/(x^4 + 2*x^2 + 1)) + 1/104*sqrt(13)*sqrt(8*I - 12)*log((sqrt(13)*sqrt(8*I - 12)*(-(54
4*I - 731)*x^4 - (1524*I + 466)*x^3 - (156*I + 1586)*x^2 + (1524*I + 466)*x - 544*I + 731) - 52*sqrt(x^4 + x^3
- x^2 - x + 1)*((140*I + 171)*x^2 - (342*I - 280)*x - 140*I - 171))/(x^4 + 2*x^2 + 1)) - 1/104*sqrt(13)*sqrt(
8*I - 12)*log((sqrt(13)*sqrt(8*I - 12)*((544*I - 731)*x^4 + (1524*I + 466)*x^3 + (156*I + 1586)*x^2 - (1524*I
+ 466)*x + 544*I - 731) - 52*sqrt(x^4 + x^3 - x^2 - x + 1)*((140*I + 171)*x^2 - (342*I - 280)*x - 140*I - 171)
)/(x^4 + 2*x^2 + 1)) - 1/104*sqrt(13)*sqrt(-8*I - 12)*log((sqrt(13)*sqrt(-8*I - 12)*(-(544*I + 731)*x^4 - (152
4*I - 466)*x^3 - (156*I - 1586)*x^2 + (1524*I - 466)*x - 544*I - 731) - 52*sqrt(x^4 + x^3 - x^2 - x + 1)*(-(14
0*I - 171)*x^2 + (342*I + 280)*x + 140*I - 171))/(x^4 + 2*x^2 + 1))

Sympy [N/A]

Not integrable

Time = 1.88 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.24 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}\, dx \]

[In]

integrate((x**2-1)/(x**2+1)/(x**4+x**3-x**2-x+1)**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)/((x**2 + 1)*sqrt(x**4 + x**3 - x**2 - x + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + 1\right )}} \,d x } \]

[In]

integrate((x^2-1)/(x^2+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^2 - 1)/(sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + 1\right )}} \,d x } \]

[In]

integrate((x^2-1)/(x^2+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="giac")

[Out]

integrate((x^2 - 1)/(sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 + 1)), x)

Mupad [N/A]

Not integrable

Time = 6.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+x^3-x^2-x+1}} \,d x \]

[In]

int((x^2 - 1)/((x^2 + 1)*(x^3 - x^2 - x + x^4 + 1)^(1/2)),x)

[Out]

int((x^2 - 1)/((x^2 + 1)*(x^3 - x^2 - x + x^4 + 1)^(1/2)), x)