\(\int \frac {(-1+x^2) (1+x^2)^3 \sqrt {1+2 x^2+x^4}}{(1+x^4) (1-x^2+x^4-x^6+x^8)} \, dx\) [2027]

Optimal result

Integrand size = 54, antiderivative size = 144 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=\frac {\sqrt {\left (1+x^2\right )^2} \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+x^2}\right )+\frac {5}{2} \text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4-\text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^2-\log (x-\text {$\#$1}) \text {$\#$1}^4+\log (x-\text {$\#$1}) \text {$\#$1}^6}{-\text {$\#$1}+2 \text {$\#$1}^3-3 \text {$\#$1}^5+4 \text {$\#$1}^7}\&\right ]\right )}{1+x^2} \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=\int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx \]

[In]

Int[((-1 + x^2)*(1 + x^2)^3*Sqrt[1 + 2*x^2 + x^4])/((1 + x^4)*(1 - x^2 + x^4 - x^6 + x^8)),x]

[Out]

((-2*I)*Sqrt[2]*Sqrt[1 + 2*x^2 + x^4]*ArcTan[(-1)^(3/4)*x])/(1 + x^2) - (2*Sqrt[2]*Sqrt[1 + 2*x^2 + x^4]*ArcTa
nh[(-1)^(3/4)*x])/(1 + x^2) - (5*Sqrt[1 + 2*x^2 + x^4]*Defer[Int][(1 - x^2 + x^4 - x^6 + x^8)^(-1), x])/(1 + x
^2) + (5*Sqrt[1 + 2*x^2 + x^4]*Defer[Int][x^2/(1 - x^2 + x^4 - x^6 + x^8), x])/(1 + x^2) - (5*Sqrt[1 + 2*x^2 +
x^4]*Defer[Int][x^4/(1 - x^2 + x^4 - x^6 + x^8), x])/(1 + x^2) + (5*Sqrt[1 + 2*x^2 + x^4]*Defer[Int][x^6/(1 -
x^2 + x^4 - x^6 + x^8), x])/(1 + x^2)

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {4 x^2 \sqrt {1+2 x^2+x^4}}{1+x^4}+\frac {\sqrt {1+2 x^2+x^4} \left (-1+2 x^2-3 x^4+4 x^6\right )}{1-x^2+x^4-x^6+x^8}\right ) \, dx \\ & = -\left (4 \int \frac {x^2 \sqrt {1+2 x^2+x^4}}{1+x^4} \, dx\right )+\int \frac {\sqrt {1+2 x^2+x^4} \left (-1+2 x^2-3 x^4+4 x^6\right )}{1-x^2+x^4-x^6+x^8} \, dx \\ & = -\left (4 \int \left (-\frac {\sqrt {1+2 x^2+x^4}}{2 \left (i-x^2\right )}+\frac {\sqrt {1+2 x^2+x^4}}{2 \left (i+x^2\right )}\right ) \, dx\right )+\int \left (\frac {\sqrt {1+2 x^2+x^4}}{-1+x^2-x^4+x^6-x^8}+\frac {2 x^2 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8}-\frac {3 x^4 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8}+\frac {4 x^6 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8}\right ) \, dx \\ & = 2 \int \frac {\sqrt {1+2 x^2+x^4}}{i-x^2} \, dx-2 \int \frac {\sqrt {1+2 x^2+x^4}}{i+x^2} \, dx+2 \int \frac {x^2 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8} \, dx-3 \int \frac {x^4 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8} \, dx+4 \int \frac {x^6 \sqrt {1+2 x^2+x^4}}{1-x^2+x^4-x^6+x^8} \, dx+\int \frac {\sqrt {1+2 x^2+x^4}}{-1+x^2-x^4+x^6-x^8} \, dx \\ & = \frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1+x^2}{i-x^2} \, dx}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1+x^2}{i+x^2} \, dx}{1+x^2}+\frac {\sqrt {1+2 x^2+x^4} \int \frac {2+2 x^2}{-1+x^2-x^4+x^6-x^8} \, dx}{2+2 x^2}+\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2 \left (2+2 x^2\right )}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (3 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4 \left (2+2 x^2\right )}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6 \left (2+2 x^2\right )}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2} \\ & = -\frac {4 x \sqrt {1+2 x^2+x^4}}{1+x^2}-\frac {\left ((2-2 i) \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{i+x^2} \, dx}{1+x^2}+\frac {\left ((2+2 i) \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{i-x^2} \, dx}{1+x^2}+\frac {\sqrt {1+2 x^2+x^4} \int \left (-\frac {2}{1-x^2+x^4-x^6+x^8}-\frac {2 x^2}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}+\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \left (\frac {2 x^2}{1-x^2+x^4-x^6+x^8}+\frac {2 x^4}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}-\frac {\left (3 \sqrt {1+2 x^2+x^4}\right ) \int \left (\frac {2 x^4}{1-x^2+x^4-x^6+x^8}+\frac {2 x^6}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \left (2-\frac {2 \left (1-x^2+x^4-2 x^6\right )}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2} \\ & = -\frac {2 i \sqrt {2} \sqrt {1+2 x^2+x^4} \arctan \left ((-1)^{3/4} x\right )}{1+x^2}-\frac {2 \sqrt {2} \sqrt {1+2 x^2+x^4} \text {arctanh}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1-x^2+x^4-2 x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2} \\ & = -\frac {2 i \sqrt {2} \sqrt {1+2 x^2+x^4} \arctan \left ((-1)^{3/4} x\right )}{1+x^2}-\frac {2 \sqrt {2} \sqrt {1+2 x^2+x^4} \text {arctanh}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \left (\frac {1}{1-x^2+x^4-x^6+x^8}-\frac {x^2}{1-x^2+x^4-x^6+x^8}+\frac {x^4}{1-x^2+x^4-x^6+x^8}-\frac {2 x^6}{1-x^2+x^4-x^6+x^8}\right ) \, dx}{2+2 x^2} \\ & = -\frac {2 i \sqrt {2} \sqrt {1+2 x^2+x^4} \arctan \left ((-1)^{3/4} x\right )}{1+x^2}-\frac {2 \sqrt {2} \sqrt {1+2 x^2+x^4} \text {arctanh}\left ((-1)^{3/4} x\right )}{1+x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (2 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (4 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (6 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^2}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}-\frac {\left (8 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^4}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2}+\frac {\left (16 \sqrt {1+2 x^2+x^4}\right ) \int \frac {x^6}{1-x^2+x^4-x^6+x^8} \, dx}{2+2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=\frac {4 \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {\left (1+x^2\right )^2}}\right )-\sqrt {5-\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} x}{\sqrt {\left (1+x^2\right )^2}}\right )-\sqrt {5+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} x}{\sqrt {\left (1+x^2\right )^2}}\right )}{\sqrt {2}} \]

[In]

Integrate[((-1 + x^2)*(1 + x^2)^3*Sqrt[1 + 2*x^2 + x^4])/((1 + x^4)*(1 - x^2 + x^4 - x^6 + x^8)),x]

[Out]

(4*ArcTanh[(Sqrt[2]*x)/Sqrt[(1 + x^2)^2]] - Sqrt[5 - Sqrt[5]]*ArcTanh[(Sqrt[(5 - Sqrt[5])/2]*x)/Sqrt[(1 + x^2)
^2]] - Sqrt[5 + Sqrt[5]]*ArcTanh[(Sqrt[(5 + Sqrt[5])/2]*x)/Sqrt[(1 + x^2)^2]])/Sqrt[2]

Maple [N/A] (verified)

Time = 0.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \sqrt {2}\, \ln \left (x \sqrt {2}+x^{2}+1\right )}{x^{2}+1}-\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \sqrt {2}\, \ln \left (-x \sqrt {2}+x^{2}+1\right )}{x^{2}+1}+\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-5 \textit {\_Z}^{2}+5\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R} x +x^{2}+1\right )\right )}{2 x^{2}+2}\)	\(110\)
pseudoelliptic	\(\frac {\left (\sqrt {10+2 \sqrt {5}}\, \left (-5+\sqrt {5}\right ) \operatorname {arctanh}\left (\frac {2 x^{2}+2}{x \sqrt {10-2 \sqrt {5}}}\right )-\sqrt {10-2 \sqrt {5}}\, \left (5+\sqrt {5}\right ) \operatorname {arctanh}\left (\frac {2 x^{2}+2}{x \sqrt {10+2 \sqrt {5}}}\right )+8 \sqrt {5}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+1\right ) \sqrt {2}}{2 x}\right )\right ) \operatorname {csgn}\left (\frac {x^{2}+1}{x}\right ) \sqrt {5}}{20}\)	\(110\)
default	\(\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \left (\sqrt {2}\, \ln \left (-\frac {x \sqrt {2}+x^{2}+1}{x \sqrt {2}-x^{2}-1}\right )-\sqrt {2}\, \ln \left (-\frac {x \sqrt {2}-x^{2}-1}{x \sqrt {2}+x^{2}+1}\right )+5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (125 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-5 \textit {\_R} x +x^{2}+1\right )\right )\right )}{2 x^{2}+2}\)	\(113\)

[In]

int((x^2-1)*(x^2+1)^3*((x^2+1)^2)^(1/2)/(x^4+1)/(x^8-x^6+x^4-x^2+1),x,method=_RETURNVERBOSE)

[Out]

((x^2+1)^2)^(1/2)/(x^2+1)*2^(1/2)*ln(x*2^(1/2)+x^2+1)-((x^2+1)^2)^(1/2)/(x^2+1)*2^(1/2)*ln(-x*2^(1/2)+x^2+1)+1
/2*((x^2+1)^2)^(1/2)/(x^2+1)*sum(_R*ln(-_R*x+x^2+1),_R=RootOf(_Z^4-5*_Z^2+5))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=-\frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (2 \, x^{2} + \sqrt {2} x \sqrt {\sqrt {5} + 5} + 2\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (2 \, x^{2} - \sqrt {2} x \sqrt {\sqrt {5} + 5} + 2\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (2 \, x^{2} + \sqrt {2} x \sqrt {-\sqrt {5} + 5} + 2\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (2 \, x^{2} - \sqrt {2} x \sqrt {-\sqrt {5} + 5} + 2\right ) + \sqrt {2} \log \left (\frac {x^{4} + 4 \, x^{2} + 2 \, \sqrt {2} {\left (x^{3} + x\right )} + 1}{x^{4} + 1}\right ) \]

[In]

integrate((x^2-1)*(x^2+1)^3*((x^2+1)^2)^(1/2)/(x^4+1)/(x^8-x^6+x^4-x^2+1),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*sqrt(sqrt(5) + 5)*log(2*x^2 + sqrt(2)*x*sqrt(sqrt(5) + 5) + 2) + 1/4*sqrt(2)*sqrt(sqrt(5) + 5)*lo
g(2*x^2 - sqrt(2)*x*sqrt(sqrt(5) + 5) + 2) - 1/4*sqrt(2)*sqrt(-sqrt(5) + 5)*log(2*x^2 + sqrt(2)*x*sqrt(-sqrt(5
) + 5) + 2) + 1/4*sqrt(2)*sqrt(-sqrt(5) + 5)*log(2*x^2 - sqrt(2)*x*sqrt(-sqrt(5) + 5) + 2) + sqrt(2)*log((x^4
+ 4*x^2 + 2*sqrt(2)*(x^3 + x) + 1)/(x^4 + 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**2-1)*(x**2+1)**3*((x**2+1)**2)**(1/2)/(x**4+1)/(x**8-x**6+x**4-x**2+1),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.49 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=\int { \frac {\sqrt {{\left (x^{2} + 1\right )}^{2}} {\left (x^{2} + 1\right )}^{3} {\left (x^{2} - 1\right )}}{{\left (x^{8} - x^{6} + x^{4} - x^{2} + 1\right )} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((x^2-1)*(x^2+1)^3*((x^2+1)^2)^(1/2)/(x^4+1)/(x^8-x^6+x^4-x^2+1),x, algorithm="maxima")

[Out]

sqrt(2)*log(x^2 + sqrt(2)*x + 1) - sqrt(2)*log(x^2 - sqrt(2)*x + 1) + 5*integrate((x^6 - x^4 + x^2 - 1)/(x^8 -
x^6 + x^4 - x^2 + 1), x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.34 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=-\sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} + \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} + \frac {2}{x} \right |}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) - \frac {1}{4} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) + \frac {1}{4} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + \frac {1}{x} \right |}\right ) \]

[In]

integrate((x^2-1)*(x^2+1)^3*((x^2+1)^2)^(1/2)/(x^4+1)/(x^8-x^6+x^4-x^2+1),x, algorithm="giac")

[Out]

-sqrt(2)*log(abs(2*x - 2*sqrt(2) + 2/x)/abs(2*x + 2*sqrt(2) + 2/x)) - 1/4*sqrt(2*sqrt(5) + 10)*log(abs(x + sqr
t(1/2*sqrt(5) + 5/2) + 1/x)) + 1/4*sqrt(2*sqrt(5) + 10)*log(abs(x - sqrt(1/2*sqrt(5) + 5/2) + 1/x)) - 1/4*sqrt
(-2*sqrt(5) + 10)*log(abs(x + sqrt(-1/2*sqrt(5) + 5/2) + 1/x)) + 1/4*sqrt(-2*sqrt(5) + 10)*log(abs(x - sqrt(-1
/2*sqrt(5) + 5/2) + 1/x))

Mupad [B] (verification not implemented)

Time = 6.17 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.40 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3 \sqrt {1+2 x^2+x^4}}{\left (1+x^4\right ) \left (1-x^2+x^4-x^6+x^8\right )} \, dx=2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {420500000\,\sqrt {2}\,x}{420500000\,x^2+420500000}\right )+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {2125000\,\sqrt {2}\,x\,\sqrt {\sqrt {5}+5}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2-4250000\,x^2-4250000}-\frac {905000\,\sqrt {2}\,\sqrt {5}\,x\,\sqrt {\sqrt {5}+5}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2-4250000\,x^2-4250000}\right )\,\sqrt {\sqrt {5}+5}}{2}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {2125000\,\sqrt {2}\,x\,\sqrt {5-\sqrt {5}}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2+4250000\,x^2+4250000}+\frac {905000\,\sqrt {2}\,\sqrt {5}\,x\,\sqrt {5-\sqrt {5}}}{1810000\,\sqrt {5}+1810000\,\sqrt {5}\,x^2+4250000\,x^2+4250000}\right )\,\sqrt {5-\sqrt {5}}}{2} \]

[In]

int(((x^2 - 1)*(x^2 + 1)^3*((x^2 + 1)^2)^(1/2))/((x^4 + 1)*(x^4 - x^2 - x^6 + x^8 + 1)),x)

[Out]

2*2^(1/2)*atanh((420500000*2^(1/2)*x)/(420500000*x^2 + 420500000)) + (2^(1/2)*atanh((2125000*2^(1/2)*x*(5^(1/2
) + 5)^(1/2))/(1810000*5^(1/2) + 1810000*5^(1/2)*x^2 - 4250000*x^2 - 4250000) - (905000*2^(1/2)*5^(1/2)*x*(5^(
1/2) + 5)^(1/2))/(1810000*5^(1/2) + 1810000*5^(1/2)*x^2 - 4250000*x^2 - 4250000))*(5^(1/2) + 5)^(1/2))/2 - (2^
(1/2)*atanh((2125000*2^(1/2)*x*(5 - 5^(1/2))^(1/2))/(1810000*5^(1/2) + 1810000*5^(1/2)*x^2 + 4250000*x^2 + 425
0000) + (905000*2^(1/2)*5^(1/2)*x*(5 - 5^(1/2))^(1/2))/(1810000*5^(1/2) + 1810000*5^(1/2)*x^2 + 4250000*x^2 +
4250000))*(5 - 5^(1/2))^(1/2))/2