\(\int \frac {\sqrt {2-x^2-4 x^4} (1+2 x^4)}{(-1+2 x^4) (-1-x^2+2 x^4)} \, dx\) [989]

Optimal result

Integrand size = 47, antiderivative size = 75 \[ \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx=\arctan \left (\frac {x \sqrt {2-x^2-4 x^4}}{-2+x^2+4 x^4}\right )-\sqrt {3} \arctan \left (\frac {\sqrt {3} x \sqrt {2-x^2-4 x^4}}{-2+x^2+4 x^4}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {2-x^2-4 x^4}}\right )+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{\sqrt {2-x^2-4 x^4}}\right ) \] Input:

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

Output:

-ArcTan[x/Sqrt[2 - x^2 - 4*x^4]] + Sqrt[3]*ArcTan[(Sqrt[3]*x)/Sqrt[2 - x^2 
 - 4*x^4]]
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.42 (sec) , antiderivative size = 475, normalized size of antiderivative = 6.33, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {7276, 2009}

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.

\(\displaystyle \int \frac {\sqrt {-4 x^4-x^2+2} \left (2 x^4+1\right )}{\left (2 x^4-1\right ) \left (2 x^4-x^2-1\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (-\frac {4 \sqrt {-4 x^4-x^2+2} x^2}{2 x^4-1}+\frac {\sqrt {-4 x^4-x^2+2}}{x^2-1}+\frac {2 \sqrt {-4 x^4-x^2+2}}{2 x^2+1}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{4} \sqrt {3 \left (11 \sqrt {33}-59\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {\left (1+4 \sqrt {2}-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+\frac {\left (1-4 \sqrt {2}-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (1-\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (-\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (-1+\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )\)

Input:

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

Output:

((1 - 4*Sqrt[2] - Sqrt[33])*EllipticF[ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], 
 (-17 + Sqrt[33])/16])/Sqrt[2*(1 + Sqrt[33])] + ((1 + 4*Sqrt[2] - Sqrt[33] 
)*EllipticF[ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16])/Sqr 
t[2*(1 + Sqrt[33])] + (Sqrt[3*(13 + 3*Sqrt[33])]*EllipticF[ArcSin[2*Sqrt[2 
/(-1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16])/4 - (Sqrt[3*(-59 + 11*Sqrt[33] 
)]*EllipticF[ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16])/4 
+ 3*Sqrt[2/(1 + Sqrt[33])]*EllipticPi[(1 - Sqrt[33])/4, ArcSin[2*Sqrt[2/(- 
1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16] - Sqrt[2/(1 + Sqrt[33])]*EllipticP 
i[-1/4*(1 - Sqrt[33])/Sqrt[2], ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + 
 Sqrt[33])/16] - Sqrt[2/(1 + Sqrt[33])]*EllipticPi[(1 - Sqrt[33])/(4*Sqrt[ 
2]), ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16] + 3*Sqrt[2/ 
(1 + Sqrt[33])]*EllipticPi[(-1 + Sqrt[33])/8, ArcSin[2*Sqrt[2/(-1 + Sqrt[3 
3])]*x], (-17 + Sqrt[33])/16]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

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]
 

Maple [A] (verified)

Time = 6.47 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81

method	result	size
elliptic	\(\frac {\left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-4 x^{4}-x^{2}+2}}{x}\right )-\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {-4 x^{4}-x^{2}+2}\, \sqrt {2}}{6 x}\right )\right ) \sqrt {2}}{2}\)	\(61\)
default	\(\arctan \left (\frac {\sqrt {-4 x^{4}-x^{2}+2}}{x}\right )+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (i \left (-1+x \right )^{2} \sqrt {2}+2 x^{2}+\frac {x}{2}-1\right ) \sqrt {3}}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \left (-\frac {i \left (1+x \right )^{2} \sqrt {2}}{2}-x^{2}+\frac {x}{4}+\frac {1}{2}\right )}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}\)	\(113\)
pseudoelliptic	\(\arctan \left (\frac {\sqrt {-4 x^{4}-x^{2}+2}}{x}\right )+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (i \left (-1+x \right )^{2} \sqrt {2}+2 x^{2}+\frac {x}{2}-1\right ) \sqrt {3}}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \left (-\frac {i \left (1+x \right )^{2} \sqrt {2}}{2}-x^{2}+\frac {x}{4}+\frac {1}{2}\right )}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}\)	\(113\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+3 x \sqrt {-4 x^{4}-x^{2}+2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{2}+1\right ) \left (-1+x \right ) \left (1+x \right )}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+x \sqrt {-4 x^{4}-x^{2}+2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{4}-1}\right )}{2}\)	\(143\)

Input:

int((-4*x^4-x^2+2)^(1/2)*(2*x^4+1)/(2*x^4-1)/(2*x^4-x^2-1),x,method=_RETUR 
NVERBOSE)
 

Output:

1/2*(2^(1/2)*arctan((-4*x^4-x^2+2)^(1/2)/x)-6^(1/2)*arctan(1/6*6^(1/2)*(-4 
*x^4-x^2+2)^(1/2)*2^(1/2)/x))*2^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-4 \, x^{4} - x^{2} + 2} x}{2 \, x^{4} + 2 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-4 \, x^{4} - x^{2} + 2} x}{2 \, x^{4} + x^{2} - 1}\right ) \] Input:

integrate((-4*x^4-x^2+2)^(1/2)*(2*x^4+1)/(2*x^4-1)/(2*x^4-x^2-1),x, algori 
thm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-4*x^4-x^2+2)^(1/2)*(2*x^4+1)/(2*x^4-1)/(2*x^4-x^2-1),x, algori 
thm="maxima")
 

Output:

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

Giac [F]

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

integrate((-4*x^4-x^2+2)^(1/2)*(2*x^4+1)/(2*x^4-1)/(2*x^4-x^2-1),x, algori 
thm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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