\(\int \frac {(1+x^4) (-1+x^2+x^4)^{3/2}}{(-1+x^4) (1+x^2-x^4-x^6+x^8)} \, dx\) [1865]

Optimal result

Integrand size = 45, antiderivative size = 128 \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=-\sqrt {\frac {1}{2} \left (3-i \sqrt {3}\right )} \arctan \left (\frac {\sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}} x}{\sqrt {-1+x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (3+i \sqrt {3}\right )} \arctan \left (\frac {\sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}} x}{\sqrt {-1+x^2+x^4}}\right )-\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \]

[Out]

-1/2*(6-2*I*3^(1/2))^(1/2)*arctan(1/2*(-6-2*I*3^(1/2))^(1/2)*x/(x^4+x^2-1)^(1/2))-1/2*(6+2*I*3^(1/2))^(1/2)*ar
ctan(1/2*(-6+2*I*3^(1/2))^(1/2)*x/(x^4+x^2-1)^(1/2))-arctanh(x/(x^4+x^2-1)^(1/2))

Rubi [F]

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

[In]

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

[Out]

(4*x*(1 + Sqrt[5] + 2*x^2))/(3*Sqrt[-1 + x^2 + x^4]) - (x*(1 + 3*x^2)*Sqrt[-1 + x^2 + x^4])/15 + (x*(11 + 3*x^
2)*Sqrt[-1 + x^2 + x^4])/15 - (4*5^(1/4)*Sqrt[(2 - (1 - Sqrt[5])*x^2)/(2 - (1 + Sqrt[5])*x^2)]*Sqrt[-2 + (1 +
Sqrt[5])*x^2]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 + (1 + Sqrt[5])*x^2]], (5 + Sqrt[5])/10])/(3*Sqrt[(
2 - (1 + Sqrt[5])*x^2)^(-1)]*Sqrt[-1 + x^2 + x^4]) - ((1 - Sqrt[5])*Sqrt[1 + Sqrt[5] + 2*x^2]*Sqrt[1 + (2*x^2)
/(1 - Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])/2])/(Sqrt[2]*(3 - Sqrt[5])*Sqrt[-1
+ x^2 + x^4]) - ((1 - Sqrt[5])*Sqrt[1 + Sqrt[5] + 2*x^2]*Sqrt[1 + (2*x^2)/(1 - Sqrt[5])]*EllipticF[ArcSin[Sqrt
[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])/2])/(Sqrt[2]*(1 + Sqrt[5])*Sqrt[-1 + x^2 + x^4]) + ((1 - 4*Sqrt[5])*Sqrt
[(2 - (1 - Sqrt[5])*x^2)/(2 - (1 + Sqrt[5])*x^2)]*Sqrt[-2 + (1 + Sqrt[5])*x^2]*EllipticF[ArcSin[(Sqrt[2]*5^(1/
4)*x)/Sqrt[-2 + (1 + Sqrt[5])*x^2]], (5 + Sqrt[5])/10])/(6*5^(3/4)*Sqrt[(2 - (1 + Sqrt[5])*x^2)^(-1)]*Sqrt[-1
+ x^2 + x^4]) - ((1 - Sqrt[5])*Sqrt[(2 - (1 - Sqrt[5])*x^2)/(2 - (1 + Sqrt[5])*x^2)]*Sqrt[-2 + (1 + Sqrt[5])*x
^2]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 + (1 + Sqrt[5])*x^2]], (5 + Sqrt[5])/10])/(4*5^(1/4)*Sqrt[(2
- (1 + Sqrt[5])*x^2)^(-1)]*Sqrt[-1 + x^2 + x^4]) - (Sqrt[(2 - (1 - Sqrt[5])*x^2)/(2 - (1 + Sqrt[5])*x^2)]*Sqrt
[-2 + (1 + Sqrt[5])*x^2]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 + (1 + Sqrt[5])*x^2]], (5 + Sqrt[5])/10]
)/(5^(1/4)*(3 - Sqrt[5])*Sqrt[(2 - (1 + Sqrt[5])*x^2)^(-1)]*Sqrt[-1 + x^2 + x^4]) + ((4 - Sqrt[5])*Sqrt[(2 - (
1 - Sqrt[5])*x^2)/(2 - (1 + Sqrt[5])*x^2)]*Sqrt[-2 + (1 + Sqrt[5])*x^2]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/S
qrt[-2 + (1 + Sqrt[5])*x^2]], (5 + Sqrt[5])/10])/(6*5^(3/4)*Sqrt[(2 - (1 + Sqrt[5])*x^2)^(-1)]*Sqrt[-1 + x^2 +
x^4]) - (Sqrt[(2 - (1 - Sqrt[5])*x^2)/(2 - (1 + Sqrt[5])*x^2)]*Sqrt[-2 + (1 + Sqrt[5])*x^2]*EllipticF[ArcSin[
(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 + (1 + Sqrt[5])*x^2]], (5 + Sqrt[5])/10])/(5^(1/4)*(1 + Sqrt[5])*Sqrt[(2 - (1 + Sq
rt[5])*x^2)^(-1)]*Sqrt[-1 + x^2 + x^4]) + ((3 + Sqrt[5])*Sqrt[(2 - (1 - Sqrt[5])*x^2)/(2 - (1 + Sqrt[5])*x^2)]
*Sqrt[-2 + (1 + Sqrt[5])*x^2]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 + (1 + Sqrt[5])*x^2]], (5 + Sqrt[5]
)/10])/(4*5^(1/4)*Sqrt[(2 - (1 + Sqrt[5])*x^2)^(-1)]*Sqrt[-1 + x^2 + x^4]) - (Sqrt[2]*Sqrt[1 + Sqrt[5] + 2*x^2
]*Sqrt[1 + (2*x^2)/(1 - Sqrt[5])]*EllipticPi[(1 - Sqrt[5])/2, ArcSin[Sqrt[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])
/2])/((1 + Sqrt[5])*Sqrt[-1 + x^2 + x^4]) + (Sqrt[2]*(2 - Sqrt[5])*Sqrt[1 + Sqrt[5] + 2*x^2]*Sqrt[1 + (2*x^2)/
(1 - Sqrt[5])]*EllipticPi[(-1 + Sqrt[5])/2, ArcSin[Sqrt[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])/2])/((3 - Sqrt[5]
)*Sqrt[-1 + x^2 + x^4]) + Defer[Int][(-1 + x^2 + x^4)^(3/2)/(1 + x^2 - x^4 - x^6 + x^8), x] + 2*Defer[Int][(x^
2*(-1 + x^2 + x^4)^(3/2))/(1 + x^2 - x^4 - x^6 + x^8), x] - 2*Defer[Int][(x^4*(-1 + x^2 + x^4)^(3/2))/(1 + x^2
- x^4 - x^6 + x^8), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-1+x^2+x^4\right )^{3/2}}{-1-x^2}+\frac {\left (-1+x^2+x^4\right )^{3/2}}{-1+x^2}+\frac {\left (1+2 x^2-2 x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}\right ) \, dx \\ & = \int \frac {\left (-1+x^2+x^4\right )^{3/2}}{-1-x^2} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{-1+x^2} \, dx+\int \frac {\left (1+2 x^2-2 x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx \\ & = -\int x^2 \sqrt {-1+x^2+x^4} \, dx-\int \left (-2-x^2\right ) \sqrt {-1+x^2+x^4} \, dx-\int \frac {\sqrt {-1+x^2+x^4}}{-1-x^2} \, dx+\int \frac {\sqrt {-1+x^2+x^4}}{-1+x^2} \, dx+\int \left (\frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}+\frac {2 x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}-\frac {2 x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}\right ) \, dx \\ & = -\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}-\frac {1}{15} \int \frac {19-2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+\frac {1}{15} \int \frac {-1+8 x^2}{\sqrt {-1+x^2+x^4}} \, dx+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx+\int \frac {x^2}{\sqrt {-1+x^2+x^4}} \, dx-\int \frac {-2-x^2}{\sqrt {-1+x^2+x^4}} \, dx+\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1+x^2+x^4}} \, dx+\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+x^2+x^4}} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx \\ & = -\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+\frac {4}{15} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+2 \left (\frac {1}{2} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx\right )+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-\frac {1}{2} \left (-3-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {1}{15} \left (20-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {\int \frac {1-\sqrt {5}+2 x^2}{\left (-1+x^2\right ) \sqrt {-1+x^2+x^4}} \, dx}{-3+\sqrt {5}}+\frac {2 \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx}{-3+\sqrt {5}}+\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {\int \frac {1-\sqrt {5}+2 x^2}{\left (-1-x^2\right ) \sqrt {-1+x^2+x^4}} \, dx}{1+\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx}{1+\sqrt {5}}+\frac {1}{15} \left (-5+4 \sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\frac {1}{2} \left (-\sqrt {6-2 i \sqrt {3}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-3-i \sqrt {3}\right )} x}{\sqrt {-1+x^2+x^4}}\right )-\sqrt {6+2 i \sqrt {3}} \arctan \left (\frac {\sqrt {\frac {1}{2} i \left (3 i+\sqrt {3}\right )} x}{\sqrt {-1+x^2+x^4}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right )\right ) \]

[In]

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

[Out]

(-(Sqrt[6 - (2*I)*Sqrt[3]]*ArcTan[(Sqrt[(-3 - I*Sqrt[3])/2]*x)/Sqrt[-1 + x^2 + x^4]]) - Sqrt[6 + (2*I)*Sqrt[3]
]*ArcTan[(Sqrt[(I/2)*(3*I + Sqrt[3])]*x)/Sqrt[-1 + x^2 + x^4]] - 2*ArcTanh[x/Sqrt[-1 + x^2 + x^4]])/2

Maple [C] (verified)

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

Time = 5.69 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {3 \left (\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-7+24 i+\textit {\_Z}^{8}+\left (-8-8 i\right ) \textit {\_Z}^{6}+\left (6+24 i\right ) \textit {\_Z}^{4}+\left (-8+56 i\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+2 i-1\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}+x^{2}-1}-i}{x}\right )}{2-14 i-\textit {\_R}^{6}+\left (6+6 i\right ) \textit {\_R}^{4}+\left (-3-12 i\right ) \textit {\_R}^{2}}\right )\right ) \left (\left (\frac {2}{3}-\frac {2 x^{4}}{3}+\left (-\frac {1}{6}-i\right ) x^{2}\right ) \sqrt {x^{4}+x^{2}-1}+\frac {2 x^{6}}{3}+\left (\frac {1}{2}+i\right ) x^{4}+\left (-1+\frac {i}{2}\right ) x^{2}-\frac {2 i}{3}\right )}{\left (x^{2}-\sqrt {x^{4}+x^{2}-1}+i\right )^{3}}\)	\(224\)
pseudoelliptic	\(\frac {3 \left (\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-7+24 i+\textit {\_Z}^{8}+\left (-8-8 i\right ) \textit {\_Z}^{6}+\left (6+24 i\right ) \textit {\_Z}^{4}+\left (-8+56 i\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+2 i-1\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}+x^{2}-1}-i}{x}\right )}{2-14 i-\textit {\_R}^{6}+\left (6+6 i\right ) \textit {\_R}^{4}+\left (-3-12 i\right ) \textit {\_R}^{2}}\right )\right ) \left (\left (\frac {2}{3}-\frac {2 x^{4}}{3}+\left (-\frac {1}{6}-i\right ) x^{2}\right ) \sqrt {x^{4}+x^{2}-1}+\frac {2 x^{6}}{3}+\left (\frac {1}{2}+i\right ) x^{4}+\left (-1+\frac {i}{2}\right ) x^{2}-\frac {2 i}{3}\right )}{\left (x^{2}-\sqrt {x^{4}+x^{2}-1}+i\right )^{3}}\)	\(224\)
trager	\(-\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right ) \ln \left (\frac {8 x^{2} \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{5}+2 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{4}+8 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{2}+6 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x -2 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3}+3 x \sqrt {x^{4}+x^{2}-1}}{4 x^{2} \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}-x^{4}+2 x^{2}+1}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \ln \left (-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{4} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{4}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) x^{4}+24 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x +4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) x^{2}+6 x \sqrt {x^{4}+x^{2}-1}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )}{4 x^{2} \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+x^{4}+x^{2}-1}\right )}{2}-\frac {\ln \left (-\frac {x^{4}+2 x \sqrt {x^{4}+x^{2}-1}+2 x^{2}-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\)	\(567\)
elliptic	\(\frac {\left (-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}\, \sqrt {3}}{4}+\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}+\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}\, \sqrt {3}}{4}-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {2}}{2}\)	\(599\)

[In]

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

[Out]

3*(arctanh(1/(x^4+x^2-1)^(1/2)*((1+I)*x^2+(-1+2*I)*x-1+I))-arctanh(1/(x^4+x^2-1)^(1/2)*((1+I)*x^2+(1-2*I)*x-1+
I))-6*sum(_R*(_R^2-1+2*I)*ln((-_R*x-x^2+(x^4+x^2-1)^(1/2)-I)/x)/(2-14*I-_R^6+(6+6*I)*_R^4-(3+12*I)*_R^2),_R=Ro
otOf(-7+24*I+_Z^8-(8+8*I)*_Z^6+(6+24*I)*_Z^4+(-8+56*I)*_Z^2)))*((2/3-2/3*x^4-(1/6+I)*x^2)*(x^4+x^2-1)^(1/2)+2/
3*x^6+(1/2+I)*x^4+(-1+1/2*I)*x^2-2/3*I)/(x^2-(x^4+x^2-1)^(1/2)+I)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (84) = 168\).

Time = 0.37 (sec) , antiderivative size = 550, normalized size of antiderivative = 4.30 \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {-3} - 3} \log \left (\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} - \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {\sqrt {-3} - 3} + 12 \, {\left (x^{5} - 2 \, x^{3} + \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {-3} - 3} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} - \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {\sqrt {-3} - 3} - 12 \, {\left (x^{5} - 2 \, x^{3} + \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {-3} - 3} \log \left (\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} + \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {-\sqrt {-3} - 3} + 12 \, {\left (x^{5} - 2 \, x^{3} - \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {-3} - 3} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} + \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {-\sqrt {-3} - 3} - 12 \, {\left (x^{5} - 2 \, x^{3} - \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} - 1} x - 1}{x^{4} - 1}\right ) \]

[In]

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

[Out]

1/8*sqrt(2)*sqrt(sqrt(-3) - 3)*log((sqrt(2)*(3*x^8 + 9*x^6 - 9*x^4 - 9*x^2 - sqrt(-3)*(x^8 - x^6 - 7*x^4 + x^2
+ 1) + 3)*sqrt(sqrt(-3) - 3) + 12*(x^5 - 2*x^3 + sqrt(-3)*(x^5 - x) - x)*sqrt(x^4 + x^2 - 1))/(x^8 - x^6 - x^
4 + x^2 + 1)) - 1/8*sqrt(2)*sqrt(sqrt(-3) - 3)*log(-(sqrt(2)*(3*x^8 + 9*x^6 - 9*x^4 - 9*x^2 - sqrt(-3)*(x^8 -
x^6 - 7*x^4 + x^2 + 1) + 3)*sqrt(sqrt(-3) - 3) - 12*(x^5 - 2*x^3 + sqrt(-3)*(x^5 - x) - x)*sqrt(x^4 + x^2 - 1)
)/(x^8 - x^6 - x^4 + x^2 + 1)) + 1/8*sqrt(2)*sqrt(-sqrt(-3) - 3)*log((sqrt(2)*(3*x^8 + 9*x^6 - 9*x^4 - 9*x^2 +
sqrt(-3)*(x^8 - x^6 - 7*x^4 + x^2 + 1) + 3)*sqrt(-sqrt(-3) - 3) + 12*(x^5 - 2*x^3 - sqrt(-3)*(x^5 - x) - x)*s
qrt(x^4 + x^2 - 1))/(x^8 - x^6 - x^4 + x^2 + 1)) - 1/8*sqrt(2)*sqrt(-sqrt(-3) - 3)*log(-(sqrt(2)*(3*x^8 + 9*x^
6 - 9*x^4 - 9*x^2 + sqrt(-3)*(x^8 - x^6 - 7*x^4 + x^2 + 1) + 3)*sqrt(-sqrt(-3) - 3) - 12*(x^5 - 2*x^3 - sqrt(-
3)*(x^5 - x) - x)*sqrt(x^4 + x^2 - 1))/(x^8 - x^6 - x^4 + x^2 + 1)) + 1/2*log(-(x^4 + 2*x^2 - 2*sqrt(x^4 + x^2
- 1)*x - 1)/(x^4 - 1))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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