\(\int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} (1+2 x^2) (-1-4 x+12 x^2-8 x^3+4 x^4)} \, dx\) [611]

Optimal result

Integrand size = 68, antiderivative size = 48 \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\frac {\text {arctanh}\left (\frac {-\sqrt {\frac {2}{3}}+\sqrt {\frac {2}{3}} x}{\sqrt {\frac {1-2 x^2}{1+2 x^2}}}\right )}{\sqrt {6}} \]

[Out]

1/6*arctanh((-1/3*6^(1/2)+1/3*x*6^(1/2))/((-2*x^2+1)/(2*x^2+1))^(1/2))*6^(1/2)

Rubi [F]

\[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx \]

[In]

Int[(-1 + 4*x - 4*x^2 + 4*x^4)/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 - 4*x + 12*x^2 - 8*x^3 + 4*x^4))
,x]

[Out]

(Sqrt[1 - 2*x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1])/(Sqrt[2]*Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2]) + (
8*Sqrt[1 - 2*x^2]*Defer[Int][x/(Sqrt[1 - 4*x^4]*(-1 - 4*x + 12*x^2 - 8*x^3 + 4*x^4)), x])/(Sqrt[(1 - 2*x^2)/(1
+ 2*x^2)]*Sqrt[1 + 2*x^2]) - (16*Sqrt[1 - 2*x^2]*Defer[Int][x^2/(Sqrt[1 - 4*x^4]*(-1 - 4*x + 12*x^2 - 8*x^3 +
4*x^4)), x])/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2]) + (8*Sqrt[1 - 2*x^2]*Defer[Int][x^3/(Sqrt[1 - 4*
x^4]*(-1 - 4*x + 12*x^2 - 8*x^3 + 4*x^4)), x])/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-2 x^2} \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {1-2 x^2} \sqrt {1+2 x^2} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {\sqrt {1-2 x^2} \int \left (\frac {1}{\sqrt {1-2 x^2} \sqrt {1+2 x^2}}+\frac {8 x \left (1-2 x+x^2\right )}{\sqrt {1-2 x^2} \sqrt {1+2 x^2} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )}\right ) \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {\sqrt {1-2 x^2} \int \frac {1}{\sqrt {1-2 x^2} \sqrt {1+2 x^2}} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \int \frac {x \left (1-2 x+x^2\right )}{\sqrt {1-2 x^2} \sqrt {1+2 x^2} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {\sqrt {1-2 x^2} \int \frac {1}{\sqrt {1-4 x^4}} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \int \frac {(-1+x)^2 x}{\sqrt {1-2 x^2} \sqrt {1+2 x^2} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {\sqrt {1-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-1\right )}{\sqrt {2} \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \int \frac {(1-x)^2 x}{\sqrt {1-4 x^4} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {\sqrt {1-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-1\right )}{\sqrt {2} \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \int \left (\frac {x}{\sqrt {1-4 x^4} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )}-\frac {2 x^2}{\sqrt {1-4 x^4} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )}+\frac {x^3}{\sqrt {1-4 x^4} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )}\right ) \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {\sqrt {1-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-1\right )}{\sqrt {2} \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \int \frac {x}{\sqrt {1-4 x^4} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \int \frac {x^3}{\sqrt {1-4 x^4} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (16 \sqrt {1-2 x^2}\right ) \int \frac {x^2}{\sqrt {1-4 x^4} \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.67 \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\frac {\text {arctanh}\left (\frac {-1+x}{\sqrt {\frac {3-6 x^2}{2+4 x^2}}}\right )}{\sqrt {6}} \]

[In]

Integrate[(-1 + 4*x - 4*x^2 + 4*x^4)/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 - 4*x + 12*x^2 - 8*x^3 + 4
*x^4)),x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.83 (sec) , antiderivative size = 169, normalized size of antiderivative = 3.52

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{4}+24 \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}\, x^{3}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{3}-24 \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}\, x^{2}+12 x \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x -12 \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{4 x^{4}-8 x^{3}+12 x^{2}-4 x -1}\right )}{12}\)	\(169\)
default	\(-\frac {\left (2 x^{2}-1\right ) \left (24 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {2}, i\right )+\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-8 \textit {\_Z}^{3}+12 \textit {\_Z}^{2}-4 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (16 \sqrt {2}\, \sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticPi}\left (x \sqrt {2}, -8 \underline {\hspace {1.25 ex}}\alpha ^{3}+18 \underline {\hspace {1.25 ex}}\alpha ^{2}-28 \underline {\hspace {1.25 ex}}\alpha +14, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{3}-32 \sqrt {2}\, \sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticPi}\left (x \sqrt {2}, -8 \underline {\hspace {1.25 ex}}\alpha ^{3}+18 \underline {\hspace {1.25 ex}}\alpha ^{2}-28 \underline {\hspace {1.25 ex}}\alpha +14, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}+48 \sqrt {2}\, \sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticPi}\left (x \sqrt {2}, -8 \underline {\hspace {1.25 ex}}\alpha ^{3}+18 \underline {\hspace {1.25 ex}}\alpha ^{2}-28 \underline {\hspace {1.25 ex}}\alpha +14, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha -16 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \operatorname {EllipticPi}\left (x \sqrt {2}, -8 \underline {\hspace {1.25 ex}}\alpha ^{3}+18 \underline {\hspace {1.25 ex}}\alpha ^{2}-28 \underline {\hspace {1.25 ex}}\alpha +14, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }-\sqrt {4}\, \operatorname {arctanh}\left (\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (4 \underline {\hspace {1.25 ex}}\alpha ^{3}-9 \underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+14 \underline {\hspace {1.25 ex}}\alpha -7\right )}{\sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {-4 x^{4}+1}}\right ) \sqrt {-4 x^{4}+1}\right )}{\sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {-4 x^{4}+1}}\right ) \sqrt {-4 x^{4}+1}\right )}{48 \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}\, \sqrt {-\left (2 x^{2}+1\right ) \left (2 x^{2}-1\right )}\, \sqrt {-4 x^{4}+1}}\)	\(505\)

[In]

int((4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(4*x^4-8*x^3+12*x^2-4*x-1),x,method=_RETURNVERB
OSE)

[Out]

-1/12*RootOf(_Z^2-6)*ln(-(-4*RootOf(_Z^2-6)*x^4+24*(-(2*x^2-1)/(2*x^2+1))^(1/2)*x^3+8*RootOf(_Z^2-6)*x^3-24*(-
(2*x^2-1)/(2*x^2+1))^(1/2)*x^2+12*x*(-(2*x^2-1)/(2*x^2+1))^(1/2)+4*RootOf(_Z^2-6)*x-12*(-(2*x^2-1)/(2*x^2+1))^
(1/2)-5*RootOf(_Z^2-6))/(4*x^4-8*x^3+12*x^2-4*x-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (39) = 78\).

Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 3.12 \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\frac {1}{24} \, \sqrt {6} \log \left (-\frac {16 \, x^{8} - 64 \, x^{7} - 32 \, x^{6} + 160 \, x^{5} + 8 \, x^{4} - 80 \, x^{3} + 40 \, x^{2} + 4 \, \sqrt {6} {\left (8 \, x^{7} - 24 \, x^{6} + 20 \, x^{5} - 20 \, x^{4} + 26 \, x^{3} - 14 \, x^{2} + 9 \, x - 5\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} - 88 \, x + 49}{16 \, x^{8} - 64 \, x^{7} + 160 \, x^{6} - 224 \, x^{5} + 200 \, x^{4} - 80 \, x^{3} - 8 \, x^{2} + 8 \, x + 1}\right ) \]

[In]

integrate((4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(4*x^4-8*x^3+12*x^2-4*x-1),x, algorithm="
fricas")

[Out]

1/24*sqrt(6)*log(-(16*x^8 - 64*x^7 - 32*x^6 + 160*x^5 + 8*x^4 - 80*x^3 + 40*x^2 + 4*sqrt(6)*(8*x^7 - 24*x^6 +
20*x^5 - 20*x^4 + 26*x^3 - 14*x^2 + 9*x - 5)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)) - 88*x + 49)/(16*x^8 - 64*x^7 + 16
0*x^6 - 224*x^5 + 200*x^4 - 80*x^3 - 8*x^2 + 8*x + 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((4*x**4-4*x**2+4*x-1)/((-2*x**2+1)/(2*x**2+1))**(1/2)/(2*x**2+1)/(4*x**4-8*x**3+12*x**2-4*x-1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\int { \frac {4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1}{{\left (4 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} - 4 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]

[In]

integrate((4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(4*x^4-8*x^3+12*x^2-4*x-1),x, algorithm="
maxima")

[Out]

integrate((4*x^4 - 4*x^2 + 4*x - 1)/((4*x^4 - 8*x^3 + 12*x^2 - 4*x - 1)*(2*x^2 + 1)*sqrt(-(2*x^2 - 1)/(2*x^2 +
1))), x)

Giac [F]

\[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\int { \frac {4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1}{{\left (4 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} - 4 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]

[In]

integrate((4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(4*x^4-8*x^3+12*x^2-4*x-1),x, algorithm="
giac")

[Out]

integrate((4*x^4 - 4*x^2 + 4*x - 1)/((4*x^4 - 8*x^3 + 12*x^2 - 4*x - 1)*(2*x^2 + 1)*sqrt(-(2*x^2 - 1)/(2*x^2 +
1))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=-\int \frac {4\,x^4-4\,x^2+4\,x-1}{\left (2\,x^2+1\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,\left (-4\,x^4+8\,x^3-12\,x^2+4\,x+1\right )} \,d x \]

[In]

int(-(4*x - 4*x^2 + 4*x^4 - 1)/((2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/2)*(4*x - 12*x^2 + 8*x^3 - 4*x^4 + 1
)),x)

[Out]

-int((4*x - 4*x^2 + 4*x^4 - 1)/((2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/2)*(4*x - 12*x^2 + 8*x^3 - 4*x^4 + 1
)), x)