\(\int \frac {\sqrt {1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx\) [868]

Optimal result

Integrand size = 25, antiderivative size = 66 \[ \int \frac {\sqrt {1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx=\frac {\sqrt {1+6 x^2+x^4}}{4 (1+x)^2}-\frac {\text {arctanh}\left (\frac {2 \sqrt {2} x}{1-2 x+x^2+\sqrt {1+6 x^2+x^4}}\right )}{2 \sqrt {2}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(x*(3 + 2*Sqrt[2] + x^2))/(4*Sqrt[1 + 6*x^2 + x^4]) - ArcTanh[(2*Sqrt[2]*x)/Sqrt[1 + 6*x^2 + x^4]]/(2*Sqrt[2])
- (Sqrt[3 + 2*Sqrt[2]]*Sqrt[(1 + (3 - 2*Sqrt[2])*x^2)/(1 + (3 + 2*Sqrt[2])*x^2)]*(1 + (3 + 2*Sqrt[2])*x^2)*El
lipticE[ArcTan[Sqrt[3 + 2*Sqrt[2]]*x], -4*(4 - 3*Sqrt[2])])/(4*Sqrt[1 + 6*x^2 + x^4]) + (3*Sqrt[(1 + (3 - 2*Sq
rt[2])*x^2)/(1 + (3 + 2*Sqrt[2])*x^2)]*(1 + (3 + 2*Sqrt[2])*x^2)*EllipticF[ArcTan[Sqrt[3 + 2*Sqrt[2]]*x], -4*(
4 - 3*Sqrt[2])])/(4*Sqrt[3 + 2*Sqrt[2]]*Sqrt[1 + 6*x^2 + x^4]) - Defer[Int][Sqrt[1 + 6*x^2 + x^4]/(1 + x)^3, x
]/2 - Defer[Int][Sqrt[1 + 6*x^2 + x^4]/(1 + x)^2, x]/4

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {1+6 x^2+x^4}}{2 (1+x)^3}-\frac {\sqrt {1+6 x^2+x^4}}{4 (1+x)^2}+\frac {\sqrt {1+6 x^2+x^4}}{4 \left (-1+x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx\right )+\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{-1+x^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx \\ & = -\left (\frac {1}{4} \int \frac {-7-x^2}{\sqrt {1+6 x^2+x^4}} \, dx\right )-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx+2 \int \frac {1}{\left (-1+x^2\right ) \sqrt {1+6 x^2+x^4}} \, dx \\ & = \frac {1}{4} \int \frac {x^2}{\sqrt {1+6 x^2+x^4}} \, dx-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx+\frac {7}{4} \int \frac {1}{\sqrt {1+6 x^2+x^4}} \, dx-\int \frac {1}{\sqrt {1+6 x^2+x^4}} \, dx-\int \frac {-1-x^2}{\left (-1+x^2\right ) \sqrt {1+6 x^2+x^4}} \, dx \\ & = \frac {x \left (3+2 \sqrt {2}+x^2\right )}{4 \sqrt {1+6 x^2+x^4}}-\frac {\sqrt {3+2 \sqrt {2}} \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) E\left (\arctan \left (\sqrt {3+2 \sqrt {2}} x\right )|-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {1+6 x^2+x^4}}+\frac {3 \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {3+2 \sqrt {2}} x\right ),-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {3+2 \sqrt {2}} \sqrt {1+6 x^2+x^4}}-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx+\text {Subst}\left (\int \frac {1}{-1+8 x^2} \, dx,x,\frac {x}{\sqrt {1+6 x^2+x^4}}\right ) \\ & = \frac {x \left (3+2 \sqrt {2}+x^2\right )}{4 \sqrt {1+6 x^2+x^4}}-\frac {\text {arctanh}\left (\frac {2 \sqrt {2} x}{\sqrt {1+6 x^2+x^4}}\right )}{2 \sqrt {2}}-\frac {\sqrt {3+2 \sqrt {2}} \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) E\left (\arctan \left (\sqrt {3+2 \sqrt {2}} x\right )|-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {1+6 x^2+x^4}}+\frac {3 \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {3+2 \sqrt {2}} x\right ),-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {3+2 \sqrt {2}} \sqrt {1+6 x^2+x^4}}-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx=\frac {\sqrt {1+6 x^2+x^4}}{4 (1+x)^2}-\frac {\text {arctanh}\left (\frac {2 \sqrt {2} x}{1-2 x+x^2+\sqrt {1+6 x^2+x^4}}\right )}{2 \sqrt {2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {\sqrt {x^{4}+6 x^{2}+1}}{4 \left (1+x \right )^{2}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1+x \right )^{2} \sqrt {2}}{2 \sqrt {x^{4}+6 x^{2}+1}}\right )}{8}\)	\(49\)
default	\(\frac {-\left (1+x \right )^{2} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1+x \right )^{2} \sqrt {2}}{2 \sqrt {x^{4}+6 x^{2}+1}}\right )+2 \sqrt {x^{4}+6 x^{2}+1}}{8 \left (1+x \right )^{2}}\)	\(56\)
pseudoelliptic	\(\frac {-\left (1+x \right )^{2} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1+x \right )^{2} \sqrt {2}}{2 \sqrt {x^{4}+6 x^{2}+1}}\right )+2 \sqrt {x^{4}+6 x^{2}+1}}{8 \left (1+x \right )^{2}}\)	\(56\)
trager	\(\frac {\sqrt {x^{4}+6 x^{2}+1}}{4 \left (1+x \right )^{2}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +2 \sqrt {x^{4}+6 x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x -1\right )^{2}}\right )}{8}\)	\(80\)
elliptic	\(\frac {{\left (\left (x^{2}-1\right )^{2}+8 x^{2}\right )}^{\frac {3}{2}}}{16 \left (x^{2}-1\right )^{2}}-\frac {{\left (\left (x^{2}-1\right )^{2}+8 x^{2}\right )}^{\frac {3}{2}}}{32 \left (x^{2}-1\right )}+\frac {\sqrt {\left (x^{2}-1\right )^{2}+8 x^{2}}}{16}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (8 x^{2}+8\right ) \sqrt {2}}{8 \sqrt {\left (x^{2}-1\right )^{2}+8 x^{2}}}\right )}{8}+\frac {\left (2 x^{2}+6\right ) \sqrt {\left (x^{2}-1\right )^{2}+8 x^{2}}}{64}+\frac {\left (-\frac {1}{4 \left (\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}-2\right )}+\frac {\ln \left (\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}-2\right )}{8}-\frac {1}{4 \left (\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}+2\right )}-\frac {\ln \left (\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}+2\right )}{8}\right ) \sqrt {2}}{2}\)	\(232\)

[In]

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

[Out]

1/4*(x^4+6*x^2+1)^(1/2)/(1+x)^2-1/8*2^(1/2)*arctanh(1/2*(1+x)^2*2^(1/2)/(x^4+6*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx=\frac {\sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \log \left (\frac {3 \, x^{4} + 4 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 18 \, x^{2} + 4 \, x + 3}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + 4 \, \sqrt {x^{4} + 6 \, x^{2} + 1}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[In]

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

[Out]

1/16*(sqrt(2)*(x^2 + 2*x + 1)*log((3*x^4 + 4*x^3 - 2*sqrt(2)*sqrt(x^4 + 6*x^2 + 1)*(x^2 + 2*x + 1) + 18*x^2 +
4*x + 3)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) + 4*sqrt(x^4 + 6*x^2 + 1))/(x^2 + 2*x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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