\(\int \frac {-1-2 x^2+2 x^4}{(2-3 x^2+x^4) \sqrt {1+x^6}} \, dx\) [454]

Optimal result

Integrand size = 34, antiderivative size = 36 \[ \int \frac {-1-2 x^2+2 x^4}{\left (2-3 x^2+x^4\right ) \sqrt {1+x^6}} \, dx=-\frac {\text {arctanh}\left (\frac {x \sqrt {1+x^6}}{\sqrt {2} \left (1-x^2+x^4\right )}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {1+x^6}}-\frac {5-4 x^2}{\left (2-3 x^2+x^4\right ) \sqrt {1+x^6}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1+x^6}} \, dx-\int \frac {5-4 x^2}{\left (2-3 x^2+x^4\right ) \sqrt {1+x^6}} \, dx \\ & = \frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}-\int \left (-\frac {6}{\left (-4+2 x^2\right ) \sqrt {1+x^6}}-\frac {2}{\left (-2+2 x^2\right ) \sqrt {1+x^6}}\right ) \, dx \\ & = \frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}+2 \int \frac {1}{\left (-2+2 x^2\right ) \sqrt {1+x^6}} \, dx+6 \int \frac {1}{\left (-4+2 x^2\right ) \sqrt {1+x^6}} \, dx \\ & = \frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}+2 \int \left (\frac {1}{4 (-1+x) \sqrt {1+x^6}}-\frac {1}{4 (1+x) \sqrt {1+x^6}}\right ) \, dx+6 \int \left (-\frac {1}{4 \sqrt {2} \left (\sqrt {2}-x\right ) \sqrt {1+x^6}}-\frac {1}{4 \sqrt {2} \left (\sqrt {2}+x\right ) \sqrt {1+x^6}}\right ) \, dx \\ & = \frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}+\frac {1}{2} \int \frac {1}{(-1+x) \sqrt {1+x^6}} \, dx-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {1+x^6}} \, dx-\frac {3 \int \frac {1}{\left (\sqrt {2}-x\right ) \sqrt {1+x^6}} \, dx}{2 \sqrt {2}}-\frac {3 \int \frac {1}{\left (\sqrt {2}+x\right ) \sqrt {1+x^6}} \, dx}{2 \sqrt {2}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 1.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{6}+1}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x^{2}-2\right ) \left (x -1\right ) \left (1+x \right )}\right )}{4}\)	\(70\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (31) = 62\).

Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.17 \[ \int \frac {-1-2 x^2+2 x^4}{\left (2-3 x^2+x^4\right ) \sqrt {1+x^6}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (-\frac {17 \, x^{8} - 6 \, x^{6} + 13 \, x^{4} - 4 \, \sqrt {2} \sqrt {x^{6} + 1} {\left (3 \, x^{5} - x^{3} + 2 \, x\right )} + 4 \, x^{2} + 4}{x^{8} - 6 \, x^{6} + 13 \, x^{4} - 12 \, x^{2} + 4}\right ) \]

[In]

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

[Out]

1/8*sqrt(2)*log(-(17*x^8 - 6*x^6 + 13*x^4 - 4*sqrt(2)*sqrt(x^6 + 1)*(3*x^5 - x^3 + 2*x) + 4*x^2 + 4)/(x^8 - 6*
x^6 + 13*x^4 - 12*x^2 + 4))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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