Integrand size = 68, antiderivative size = 48 \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\frac {\text {arctanh}\left (\frac {-\sqrt {\frac {2}{3}}+\sqrt {\frac {2}{3}} x}{\sqrt {\frac {1-2 x^2}{1+2 x^2}}}\right )}{\sqrt {6}} \]
Time = 10.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.67 \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\frac {\text {arctanh}\left (\frac {-1+x}{\sqrt {\frac {3-6 x^2}{2+4 x^2}}}\right )}{\sqrt {6}} \]
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]
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.
| \(\displaystyle \int \frac {4 x^4-4 x^2+4 x-1}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \left (2 x^2+1\right ) \left (4 x^4-8 x^3+12 x^2-4 x-1\right )} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {1-2 x^2} \int \frac {-4 x^4+4 x^2-4 x+1}{\sqrt {1-2 x^2} \sqrt {2 x^2+1} \left (-4 x^4+8 x^3-12 x^2+4 x+1\right )}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\sqrt {1-2 x^2} \int \frac {-4 x^4+4 x^2-4 x+1}{\sqrt {1-4 x^4} \left (-4 x^4+8 x^3-12 x^2+4 x+1\right )}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {1-2 x^2} \int \left (\frac {1}{\sqrt {1-4 x^4}}-\frac {8 x \left (x^2-2 x+1\right )}{\sqrt {1-4 x^4} \left (-4 x^4+8 x^3-12 x^2+4 x+1\right )}\right )dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-2 x^2} \left (8 \int \frac {x}{\sqrt {1-4 x^4} \left (4 x^4-8 x^3+12 x^2-4 x-1\right )}dx-16 \int \frac {x^2}{\sqrt {1-4 x^4} \left (4 x^4-8 x^3+12 x^2-4 x-1\right )}dx+8 \int \frac {x^3}{\sqrt {1-4 x^4} \left (4 x^4-8 x^3+12 x^2-4 x-1\right )}dx+\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-1\right )}{\sqrt {2}}\right )}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
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]
3.7.11.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
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}}\) | \(505\) |
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=_RETURNVERBOSE)
-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))
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 \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\frac {1}{24} \, \sqrt {6} \log \left (-\frac {16 \, x^{8} - 64 \, x^{7} - 32 \, x^{6} + 160 \, x^{5} + 8 \, x^{4} - 80 \, x^{3} + 40 \, x^{2} + 4 \, \sqrt {6} {\left (8 \, x^{7} - 24 \, x^{6} + 20 \, x^{5} - 20 \, x^{4} + 26 \, x^{3} - 14 \, x^{2} + 9 \, x - 5\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} - 88 \, x + 49}{16 \, x^{8} - 64 \, x^{7} + 160 \, x^{6} - 224 \, x^{5} + 200 \, x^{4} - 80 \, x^{3} - 8 \, x^{2} + 8 \, x + 1}\right ) \]
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")
1/24*sqrt(6)*log(-(16*x^8 - 64*x^7 - 32*x^6 + 160*x^5 + 8*x^4 - 80*x^3 + 4 0*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 + 160* x^6 - 224*x^5 + 200*x^4 - 80*x^3 - 8*x^2 + 8*x + 1))
Timed out. \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\text {Timed out} \]
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)
\[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\int { \frac {4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1}{{\left (4 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} - 4 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]
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")
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)
\[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=\int { \frac {4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1}{{\left (4 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} - 4 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]
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")
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)
Timed out. \[ \int \frac {-1+4 x-4 x^2+4 x^4}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-4 x+12 x^2-8 x^3+4 x^4\right )} \, dx=-\int \frac {4\,x^4-4\,x^2+4\,x-1}{\left (2\,x^2+1\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,\left (-4\,x^4+8\,x^3-12\,x^2+4\,x+1\right )} \,d x \]
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)