\(\int \frac {-1-2 x^2+2 x^4}{x^2 (1+x^2) \sqrt {1+x^6}} \, dx\) [183]

Optimal result

Integrand size = 32, antiderivative size = 20 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{x \left (1+x^2\right )} \]

[Out]

(x^6+1)^(1/2)/x/(x^2+1)

Rubi [F]

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

[In]

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

[Out]

Sqrt[1 + x^6]/x - ((1 + Sqrt[3])*x*Sqrt[1 + x^6])/(1 + (1 + Sqrt[3])*x^2) + (3^(1/4)*x*(1 + x^2)*Sqrt[(1 - x^2
+ x^4)/(1 + (1 + Sqrt[3])*x^2)^2]*EllipticE[ArcCos[(1 + (1 - Sqrt[3])*x^2)/(1 + (1 + Sqrt[3])*x^2)], (2 + Sqr
t[3])/4])/(Sqrt[(x^2*(1 + x^2))/(1 + (1 + Sqrt[3])*x^2)^2]*Sqrt[1 + x^6]) + (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]) + ((1 - Sqrt[3])*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])/(2*3^(1/4)*Sqrt[(x^2*(1 + x^2))/(1 + (1 + Sqrt[3])*x^2)^2]*Sqrt[1 + x^6]) - ((3*I)/2)*Defer[In
t][1/((I - x)*Sqrt[1 + x^6]), x] - ((3*I)/2)*Defer[Int][1/((I + x)*Sqrt[1 + x^6]), x]

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

Mathematica [A] (verified)

Time = 8.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{x+x^3} \]

[In]

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

[Out]

Sqrt[1 + x^6]/(x + x^3)

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
trager	\(\frac {\sqrt {x^{6}+1}}{x \left (x^{2}+1\right )}\)	\(19\)
gosper	\(\frac {x^{4}-x^{2}+1}{x \sqrt {x^{6}+1}}\)	\(22\)
risch	\(\frac {x^{4}-x^{2}+1}{x \sqrt {x^{6}+1}}\)	\(22\)

[In]

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

[Out]

(x^6+1)^(1/2)/x/(x^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{x^{3} + x} \]

[In]

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

[Out]

sqrt(x^6 + 1)/(x^3 + x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{4} - x^{2} + 1}}{\sqrt {x^{2} + 1} x} \]

[In]

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

[Out]

sqrt(x^4 - x^2 + 1)/(sqrt(x^2 + 1)*x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^6+1}}{x\,\left (x^2+1\right )} \]

[In]

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

[Out]

(x^6 + 1)^(1/2)/(x*(x^2 + 1))