\(\int \frac {(1+x^4) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx\) [1292]

Optimal result

Integrand size = 40, antiderivative size = 93 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\frac {1}{10} \left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {-1-x^2+x^4}}\right )+\frac {1}{10} \left (5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {-1-x^2+x^4}}\right ) \]

[Out]

1/10*(-5-5^(1/2))*arctan((1/2+1/2*5^(1/2))*x/(x^4-x^2-1)^(1/2))+1/10*(5-5^(1/2))*arctan(2^(1/2)/(3+5^(1/2))^(1
/2)*x/(x^4-x^2-1)^(1/2))

Rubi [F]

\[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx \]

[In]

Int[((1 + x^4)*Sqrt[-1 - x^2 + x^4])/(1 - x^2 - 3*x^4 + x^6 + x^8),x]

[Out]

Defer[Int][Sqrt[-1 - x^2 + x^4]/(1 - x^2 - 3*x^4 + x^6 + x^8), x] + Defer[Int][(x^4*Sqrt[-1 - x^2 + x^4])/(1 -
x^2 - 3*x^4 + x^6 + x^8), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8}+\frac {x^4 \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8}\right ) \, dx \\ & = \int \frac {\sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx+\int \frac {x^4 \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\frac {1}{10} \left (-\left (\left (-5+\sqrt {5}\right ) \arctan \left (\frac {\left (-1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right )\right )-\left (5+\sqrt {5}\right ) \arctan \left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right )\right ) \]

[In]

Integrate[((1 + x^4)*Sqrt[-1 - x^2 + x^4])/(1 - x^2 - 3*x^4 + x^6 + x^8),x]

[Out]

(-((-5 + Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^4])]) - (5 + Sqrt[5])*ArcTan[((1 + Sqrt[5])*x
)/(2*Sqrt[-1 - x^2 + x^4])])/10

Maple [A] (verified)

Time = 3.74 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30

method	result	size
default	\(\frac {\left (\frac {4 \left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right ) \sqrt {2}}{2}\)	\(121\)
elliptic	\(\frac {\left (\frac {4 \left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right ) \sqrt {2}}{2}\)	\(121\)
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-3+4 i+\textit {\_Z}^{4}+2 i \textit {\_Z}^{3}+\left (2-4 i\right ) \textit {\_Z}^{2}+\left (4-2 i\right ) \textit {\_Z} \right )}{\sum }\frac {\left (1+2 i+i \textit {\_R}^{3}-\textit {\_R}^{2}+\left (2+i\right ) \textit {\_R} \right ) \ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}-x^{2}-1}-i}{x}\right )}{2 \textit {\_R}^{3}+3 i \textit {\_R}^{2}+\left (2-4 i\right ) \textit {\_R} +2-i}\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-3+4 i+\textit {\_Z}^{4}-2 i \textit {\_Z}^{3}+\left (2-4 i\right ) \textit {\_Z}^{2}+\left (-4+2 i\right ) \textit {\_Z} \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}-x^{2}-1}-i}{x}\right ) \left (i \textit {\_R}^{3}+\textit {\_R}^{2}+\left (2+i\right ) \textit {\_R} -1-2 i\right )}{-2 \textit {\_R}^{3}+3 i \textit {\_R}^{2}+\left (-2+4 i\right ) \textit {\_R} +2-i}\right )}{2}\)	\(206\)
trager	\(20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {1400 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}-70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+460 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+30 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+36 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+4 x \sqrt {x^{4}-x^{2}-1}+9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )+\operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-1200 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}-60 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+60 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +60 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+x \sqrt {x^{4}-x^{2}-1}+2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}-x^{4}+x^{2}+1}\right )+3 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {1400 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}-70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+460 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+30 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+36 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+4 x \sqrt {x^{4}-x^{2}-1}+9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )\)	\(672\)

[In]

int((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(4/5*(5^(1/2)-3)*5^(1/2)/(2*10^(1/2)-2*2^(1/2))*arctan(4*(x^4-x^2-1)^(1/2)*2^(1/2)/x/(2*10^(1/2)-2*2^(1/2)
))+4/5*(3+5^(1/2))*5^(1/2)/(2*10^(1/2)+2*2^(1/2))*arctan(4*(x^4-x^2-1)^(1/2)*2^(1/2)/x/(2*10^(1/2)+2*2^(1/2)))
)*2^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (61) = 122\).

Time = 0.35 (sec) , antiderivative size = 575, normalized size of antiderivative = 6.18 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=-\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 3} \log \left (\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} - \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {\sqrt {5} - 3} + 20 \, {\left (7 \, x^{5} - 4 \, x^{3} - \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 3} \log \left (-\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} - \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {\sqrt {5} - 3} - 20 \, {\left (7 \, x^{5} - 4 \, x^{3} - \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 3} \log \left (\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {-\sqrt {5} - 3} + 20 \, {\left (7 \, x^{5} - 4 \, x^{3} + \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 3} \log \left (-\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {-\sqrt {5} - 3} - 20 \, {\left (7 \, x^{5} - 4 \, x^{3} + \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) \]

[In]

integrate((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x, algorithm="fricas")

[Out]

-1/40*sqrt(10)*sqrt(sqrt(5) - 3)*log((sqrt(10)*(5*x^8 - 25*x^6 + 5*x^4 + 25*x^2 - sqrt(5)*(3*x^8 - 11*x^6 - x^
4 + 11*x^2 + 3) + 5)*sqrt(sqrt(5) - 3) + 20*(7*x^5 - 4*x^3 - sqrt(5)*(3*x^5 - 2*x^3 - 3*x) - 7*x)*sqrt(x^4 - x
^2 - 1))/(x^8 + x^6 - 3*x^4 - x^2 + 1)) + 1/40*sqrt(10)*sqrt(sqrt(5) - 3)*log(-(sqrt(10)*(5*x^8 - 25*x^6 + 5*x
^4 + 25*x^2 - sqrt(5)*(3*x^8 - 11*x^6 - x^4 + 11*x^2 + 3) + 5)*sqrt(sqrt(5) - 3) - 20*(7*x^5 - 4*x^3 - sqrt(5)
*(3*x^5 - 2*x^3 - 3*x) - 7*x)*sqrt(x^4 - x^2 - 1))/(x^8 + x^6 - 3*x^4 - x^2 + 1)) - 1/40*sqrt(10)*sqrt(-sqrt(5
) - 3)*log((sqrt(10)*(5*x^8 - 25*x^6 + 5*x^4 + 25*x^2 + sqrt(5)*(3*x^8 - 11*x^6 - x^4 + 11*x^2 + 3) + 5)*sqrt(
-sqrt(5) - 3) + 20*(7*x^5 - 4*x^3 + sqrt(5)*(3*x^5 - 2*x^3 - 3*x) - 7*x)*sqrt(x^4 - x^2 - 1))/(x^8 + x^6 - 3*x
^4 - x^2 + 1)) + 1/40*sqrt(10)*sqrt(-sqrt(5) - 3)*log(-(sqrt(10)*(5*x^8 - 25*x^6 + 5*x^4 + 25*x^2 + sqrt(5)*(3
*x^8 - 11*x^6 - x^4 + 11*x^2 + 3) + 5)*sqrt(-sqrt(5) - 3) - 20*(7*x^5 - 4*x^3 + sqrt(5)*(3*x^5 - 2*x^3 - 3*x)
- 7*x)*sqrt(x^4 - x^2 - 1))/(x^8 + x^6 - 3*x^4 - x^2 + 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\text {Timed out} \]

[In]

integrate((x**4+1)*(x**4-x**2-1)**(1/2)/(x**8+x**6-3*x**4-x**2+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int { \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1} \,d x } \]

[In]

integrate((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 - x^2 - 1)*(x^4 + 1)/(x^8 + x^6 - 3*x^4 - x^2 + 1), x)

Giac [F]

\[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int { \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1} \,d x } \]

[In]

integrate((x^4+1)*(x^4-x^2-1)^(1/2)/(x^8+x^6-3*x^4-x^2+1),x, algorithm="giac")

[Out]

integrate(sqrt(x^4 - x^2 - 1)*(x^4 + 1)/(x^8 + x^6 - 3*x^4 - x^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int \frac {\left (x^4+1\right )\,\sqrt {x^4-x^2-1}}{x^8+x^6-3\,x^4-x^2+1} \,d x \]

[In]

int(((x^4 + 1)*(x^4 - x^2 - 1)^(1/2))/(x^6 - 3*x^4 - x^2 + x^8 + 1),x)

[Out]

int(((x^4 + 1)*(x^4 - x^2 - 1)^(1/2))/(x^6 - 3*x^4 - x^2 + x^8 + 1), x)