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

Optimal. Leaf size=128 \[ -\sqrt {\frac {1}{2} \left (3-i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}} x}{\sqrt {x^4+x^2-1}}\right )-\sqrt {\frac {1}{2} \left (3+i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}} x}{\sqrt {x^4+x^2-1}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2-1}}\right ) \]

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Rubi [F]  time = 2.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\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 {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+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 {\left (\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {\sqrt {1-\sqrt {5}+2 x^2}}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right )} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {\sqrt {1-\sqrt {5}+2 x^2}}{\left (-1-x^2\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\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 {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+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 {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (3-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\frac {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (-1-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\left (-1-x^2\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\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 {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+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 {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (3-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\frac {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (-1-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\left (-1-x^2\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\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 {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )-\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{\sqrt {2} \left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} \Pi \left (\frac {1}{2} \left (1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{2 \sqrt {2} \sqrt {-1+x^2+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} \Pi \left (\frac {1}{2} \left (-1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{2 \sqrt {2} \sqrt {-1+x^2+x^4}}+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 {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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]

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

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IntegrateAlgebraic [A]  time = 1.32, size = 128, normalized size = 1.00 \begin {gather*} -\sqrt {\frac {1}{2} \left (3-i \sqrt {3}\right )} \tan ^{-1}\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 )} \tan ^{-1}\left (\frac {\sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}} x}{\sqrt {-1+x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 4.27, size = 4669, normalized size = 36.48

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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]

-1/16*12^(1/4)*(sqrt(3)*sqrt(2) - 2*sqrt(2))*sqrt(sqrt(3) + 2)*log(48*(12*x^6 + 12*x^4 + 12^(1/4)*sqrt(x^4 + x
^2 - 1)*(3*sqrt(2)*x^3 + sqrt(3)*sqrt(2)*(x^5 + x^3 - x))*sqrt(sqrt(3) + 2) - 12*x^2 + sqrt(3)*(x^8 + 5*x^6 +
5*x^4 - 5*x^2 + 1))/(x^8 - x^6 - x^4 + x^2 + 1)) + 1/16*12^(1/4)*(sqrt(3)*sqrt(2) - 2*sqrt(2))*sqrt(sqrt(3) +
2)*log(48*(12*x^6 + 12*x^4 - 12^(1/4)*sqrt(x^4 + x^2 - 1)*(3*sqrt(2)*x^3 + sqrt(3)*sqrt(2)*(x^5 + x^3 - x))*sq
rt(sqrt(3) + 2) - 12*x^2 + sqrt(3)*(x^8 + 5*x^6 + 5*x^4 - 5*x^2 + 1))/(x^8 - x^6 - x^4 + x^2 + 1)) + 1/4*12^(1
/4)*sqrt(2)*sqrt(sqrt(3) + 2)*arctan(-1/36*(36*x^48 + 648*x^46 + 108*x^44 - 26208*x^42 - 80784*x^40 + 122472*x
^38 + 679176*x^36 - 158760*x^34 - 2555388*x^32 - 118872*x^30 + 5525496*x^28 + 511704*x^26 - 7137036*x^24 - 511
704*x^22 + 5525496*x^20 + 118872*x^18 - 2555388*x^16 + 158760*x^14 + 679176*x^12 - 122472*x^10 - 80784*x^8 + 2
6208*x^6 + 108*x^4 - 648*x^2 + 6*sqrt(x^4 + x^2 - 1)*(12^(3/4)*(sqrt(3)*sqrt(2)*(2*x^45 + 37*x^43 + 31*x^41 -
895*x^39 - 1651*x^37 + 6780*x^35 + 13803*x^33 - 25689*x^31 - 50211*x^29 + 55250*x^27 + 92893*x^25 - 70967*x^23
- 92893*x^21 + 55250*x^19 + 50211*x^17 - 25689*x^15 - 13803*x^13 + 6780*x^11 + 1651*x^9 - 895*x^7 - 31*x^5 +
37*x^3 - 2*x) - 3*sqrt(2)*(x^45 + 20*x^43 + 35*x^41 - 323*x^39 - 542*x^37 + 2868*x^35 + 4170*x^33 - 13119*x^31
- 16182*x^29 + 31720*x^27 + 31130*x^25 - 42331*x^23 - 31130*x^21 + 31720*x^19 + 16182*x^17 - 13119*x^15 - 417
0*x^13 + 2868*x^11 + 542*x^9 - 323*x^7 - 35*x^5 + 20*x^3 - x)) + 36*12^(1/4)*(sqrt(3)*sqrt(2)*(2*x^43 + 37*x^4
1 + 60*x^39 - 464*x^37 - 953*x^35 + 2094*x^33 + 4501*x^31 - 5092*x^29 - 10456*x^27 + 7721*x^25 + 13692*x^23 -
7721*x^21 - 10456*x^19 + 5092*x^17 + 4501*x^15 - 2094*x^13 - 953*x^11 + 464*x^9 + 60*x^7 - 37*x^5 + 2*x^3) - 3
*sqrt(2)*(x^43 + 21*x^41 + 55*x^39 - 205*x^37 - 667*x^35 + 728*x^33 + 2835*x^31 - 1397*x^29 - 6231*x^27 + 1797
*x^25 + 8014*x^23 - 1797*x^21 - 6231*x^19 + 1397*x^17 + 2835*x^15 - 728*x^13 - 667*x^11 + 205*x^9 + 55*x^7 - 2
1*x^5 + x^3)))*sqrt(sqrt(3) + 2) - sqrt(3)*(24*(216*x^41 + 1944*x^39 + 4536*x^37 - 6696*x^35 - 35640*x^33 - 38
88*x^31 + 105192*x^29 + 44712*x^27 - 171072*x^25 - 72144*x^23 + 171072*x^21 + 44712*x^19 - 105192*x^17 - 3888*
x^15 + 35640*x^13 - 6696*x^11 - 4536*x^9 + 1944*x^7 - 216*x^5 + sqrt(3)*(6*x^43 + 72*x^41 + 858*x^39 + 3294*x^
37 - 1152*x^35 - 23760*x^33 - 14400*x^31 + 68598*x^29 + 54414*x^27 - 110712*x^25 - 79452*x^23 + 110712*x^21 +
54414*x^19 - 68598*x^17 - 14400*x^15 + 23760*x^13 - 1152*x^11 - 3294*x^9 + 858*x^7 - 72*x^5 + 6*x^3 - sqrt(3)*
(x^45 + 13*x^43 - 62*x^41 - 832*x^39 - 1721*x^37 + 3372*x^35 + 12831*x^33 - 2676*x^31 - 37065*x^29 - 6848*x^27
+ 59641*x^25 + 13942*x^23 - 59641*x^21 - 6848*x^19 + 37065*x^17 - 2676*x^15 - 12831*x^13 + 3372*x^11 + 1721*x
^9 - 832*x^7 + 62*x^5 + 13*x^3 - x)) - 18*sqrt(3)*(x^43 + 10*x^41 - 67*x^39 - 355*x^37 + 176*x^35 + 2158*x^33
+ 400*x^31 - 6023*x^29 - 2137*x^27 + 9674*x^25 + 3254*x^23 - 9674*x^21 - 2137*x^19 + 6023*x^17 + 400*x^15 - 21
58*x^13 + 176*x^11 + 355*x^9 - 67*x^7 - 10*x^5 + x^3))*sqrt(x^4 + x^2 - 1) + (12^(3/4)*(sqrt(3)*sqrt(2)*(x^48
+ 18*x^46 - 3*x^44 + 58*x^42 + 3342*x^40 + 8712*x^38 - 18214*x^36 - 67788*x^34 + 36675*x^32 + 214984*x^30 - 35
874*x^28 - 367752*x^26 + 28147*x^24 + 367752*x^22 - 35874*x^20 - 214984*x^18 + 36675*x^16 + 67788*x^14 - 18214
*x^12 - 8712*x^10 + 3342*x^8 - 58*x^6 - 3*x^4 - 18*x^2 + 1) - 3*sqrt(2)*(x^48 + 16*x^46 - 31*x^44 - 696*x^42 -
1640*x^40 + 1732*x^38 + 9026*x^36 + 3108*x^34 - 16143*x^32 - 16316*x^30 + 9658*x^28 + 27416*x^26 - 1743*x^24
- 27416*x^22 + 9658*x^20 + 16316*x^18 - 16143*x^16 - 3108*x^14 + 9026*x^12 - 1732*x^10 - 1640*x^8 + 696*x^6 -
31*x^4 - 16*x^2 + 1)) + 36*12^(1/4)*(sqrt(3)*sqrt(2)*(x^46 + 21*x^44 + 37*x^42 - 36*x^40 + 376*x^38 + 666*x^36
- 3516*x^34 - 4890*x^32 + 10920*x^30 + 13636*x^28 - 18156*x^26 - 18794*x^24 + 18156*x^22 + 13636*x^20 - 10920
*x^18 - 4890*x^16 + 3516*x^14 + 666*x^12 - 376*x^10 - 36*x^8 - 37*x^6 + 21*x^4 - x^2) - 3*sqrt(2)*(x^46 + 15*x
^44 - 29*x^42 - 438*x^40 - 338*x^38 + 2664*x^36 + 3204*x^34 - 7680*x^32 - 10338*x^30 + 13366*x^28 + 17610*x^26
- 15854*x^24 - 17610*x^22 + 13366*x^20 + 10338*x^18 - 7680*x^16 - 3204*x^14 + 2664*x^12 + 338*x^10 - 438*x^8
+ 29*x^6 + 15*x^4 - x^2)))*sqrt(sqrt(3) + 2))*sqrt((12*x^6 + 12*x^4 - 12^(1/4)*sqrt(x^4 + x^2 - 1)*(3*sqrt(2)*
x^3 + sqrt(3)*sqrt(2)*(x^5 + x^3 - x))*sqrt(sqrt(3) + 2) - 12*x^2 + sqrt(3)*(x^8 + 5*x^6 + 5*x^4 - 5*x^2 + 1))
/(x^8 - x^6 - x^4 + x^2 + 1)) + 72*sqrt(3)*(3*x^46 + 51*x^44 + 21*x^42 - 1284*x^40 - 2328*x^38 + 8136*x^36 + 1
8288*x^34 - 23832*x^32 - 60048*x^30 + 41052*x^28 + 104802*x^26 - 48246*x^24 - 104802*x^22 + 41052*x^20 + 60048
*x^18 - 23832*x^16 - 18288*x^14 + 8136*x^12 + 2328*x^10 - 1284*x^8 - 21*x^6 + 51*x^4 - 3*x^2 - sqrt(3)*(x^46 +
25*x^44 + 103*x^42 - 82*x^40 - 908*x^38 - 18*x^36 + 3858*x^34 + 522*x^32 - 9636*x^30 - 1046*x^28 + 15022*x^26
+ 1198*x^24 - 15022*x^22 - 1046*x^20 + 9636*x^18 + 522*x^16 - 3858*x^14 - 18*x^12 + 908*x^10 - 82*x^8 - 103*x
^6 + 25*x^4 - x^2)) - 72*sqrt(3)*(x^46 + 35*x^44 + 383*x^42 - 14*x^40 - 4664*x^38 - 1596*x^36 + 25464*x^34 + 1
0866*x^32 - 74448*x^30 - 30740*x^28 + 125294*x^26 + 42896*x^24 - 125294*x^22 - 30740*x^20 + 74448*x^18 + 10866
*x^16 - 25464*x^14 - 1596*x^12 + 4664*x^10 - 14*x^8 - 383*x^6 + 35*x^4 - x^2) + 36)/(x^48 + 18*x^46 - 111*x^44
- 2552*x^42 - 3606*x^40 + 27594*x^38 + 53426*x^36 - 113958*x^34 - 252837*x^32 + 250858*x^30 + 592002*x^28 - 3
53226*x^26 - 777749*x^24 + 353226*x^22 + 592002*x^20 - 250858*x^18 - 252837*x^16 + 113958*x^14 + 53426*x^12 -
27594*x^10 - 3606*x^8 + 2552*x^6 - 111*x^4 - 18*x^2 + 1)) + 1/4*12^(1/4)*sqrt(2)*sqrt(sqrt(3) + 2)*arctan(1/36
*(36*x^48 + 648*x^46 + 108*x^44 - 26208*x^42 - 80784*x^40 + 122472*x^38 + 679176*x^36 - 158760*x^34 - 2555388*
x^32 - 118872*x^30 + 5525496*x^28 + 511704*x^26 - 7137036*x^24 - 511704*x^22 + 5525496*x^20 + 118872*x^18 - 25
55388*x^16 + 158760*x^14 + 679176*x^12 - 122472*x^10 - 80784*x^8 + 26208*x^6 + 108*x^4 - 648*x^2 - 6*sqrt(x^4
+ x^2 - 1)*(12^(3/4)*(sqrt(3)*sqrt(2)*(2*x^45 + 37*x^43 + 31*x^41 - 895*x^39 - 1651*x^37 + 6780*x^35 + 13803*x
^33 - 25689*x^31 - 50211*x^29 + 55250*x^27 + 92893*x^25 - 70967*x^23 - 92893*x^21 + 55250*x^19 + 50211*x^17 -
25689*x^15 - 13803*x^13 + 6780*x^11 + 1651*x^9 - 895*x^7 - 31*x^5 + 37*x^3 - 2*x) - 3*sqrt(2)*(x^45 + 20*x^43
+ 35*x^41 - 323*x^39 - 542*x^37 + 2868*x^35 + 4170*x^33 - 13119*x^31 - 16182*x^29 + 31720*x^27 + 31130*x^25 -
42331*x^23 - 31130*x^21 + 31720*x^19 + 16182*x^17 - 13119*x^15 - 4170*x^13 + 2868*x^11 + 542*x^9 - 323*x^7 - 3
5*x^5 + 20*x^3 - x)) + 36*12^(1/4)*(sqrt(3)*sqrt(2)*(2*x^43 + 37*x^41 + 60*x^39 - 464*x^37 - 953*x^35 + 2094*x
^33 + 4501*x^31 - 5092*x^29 - 10456*x^27 + 7721*x^25 + 13692*x^23 - 7721*x^21 - 10456*x^19 + 5092*x^17 + 4501*
x^15 - 2094*x^13 - 953*x^11 + 464*x^9 + 60*x^7 - 37*x^5 + 2*x^3) - 3*sqrt(2)*(x^43 + 21*x^41 + 55*x^39 - 205*x
^37 - 667*x^35 + 728*x^33 + 2835*x^31 - 1397*x^29 - 6231*x^27 + 1797*x^25 + 8014*x^23 - 1797*x^21 - 6231*x^19
+ 1397*x^17 + 2835*x^15 - 728*x^13 - 667*x^11 + 205*x^9 + 55*x^7 - 21*x^5 + x^3)))*sqrt(sqrt(3) + 2) - sqrt(3)
*(24*(216*x^41 + 1944*x^39 + 4536*x^37 - 6696*x^35 - 35640*x^33 - 3888*x^31 + 105192*x^29 + 44712*x^27 - 17107
2*x^25 - 72144*x^23 + 171072*x^21 + 44712*x^19 - 105192*x^17 - 3888*x^15 + 35640*x^13 - 6696*x^11 - 4536*x^9 +
1944*x^7 - 216*x^5 + sqrt(3)*(6*x^43 + 72*x^41 + 858*x^39 + 3294*x^37 - 1152*x^35 - 23760*x^33 - 14400*x^31 +
68598*x^29 + 54414*x^27 - 110712*x^25 - 79452*x^23 + 110712*x^21 + 54414*x^19 - 68598*x^17 - 14400*x^15 + 237
60*x^13 - 1152*x^11 - 3294*x^9 + 858*x^7 - 72*x^5 + 6*x^3 - sqrt(3)*(x^45 + 13*x^43 - 62*x^41 - 832*x^39 - 172
1*x^37 + 3372*x^35 + 12831*x^33 - 2676*x^31 - 37065*x^29 - 6848*x^27 + 59641*x^25 + 13942*x^23 - 59641*x^21 -
6848*x^19 + 37065*x^17 - 2676*x^15 - 12831*x^13 + 3372*x^11 + 1721*x^9 - 832*x^7 + 62*x^5 + 13*x^3 - x)) - 18*
sqrt(3)*(x^43 + 10*x^41 - 67*x^39 - 355*x^37 + 176*x^35 + 2158*x^33 + 400*x^31 - 6023*x^29 - 2137*x^27 + 9674*
x^25 + 3254*x^23 - 9674*x^21 - 2137*x^19 + 6023*x^17 + 400*x^15 - 2158*x^13 + 176*x^11 + 355*x^9 - 67*x^7 - 10
*x^5 + x^3))*sqrt(x^4 + x^2 - 1) - (12^(3/4)*(sqrt(3)*sqrt(2)*(x^48 + 18*x^46 - 3*x^44 + 58*x^42 + 3342*x^40 +
8712*x^38 - 18214*x^36 - 67788*x^34 + 36675*x^32 + 214984*x^30 - 35874*x^28 - 367752*x^26 + 28147*x^24 + 3677
52*x^22 - 35874*x^20 - 214984*x^18 + 36675*x^16 + 67788*x^14 - 18214*x^12 - 8712*x^10 + 3342*x^8 - 58*x^6 - 3*
x^4 - 18*x^2 + 1) - 3*sqrt(2)*(x^48 + 16*x^46 - 31*x^44 - 696*x^42 - 1640*x^40 + 1732*x^38 + 9026*x^36 + 3108*
x^34 - 16143*x^32 - 16316*x^30 + 9658*x^28 + 27416*x^26 - 1743*x^24 - 27416*x^22 + 9658*x^20 + 16316*x^18 - 16
143*x^16 - 3108*x^14 + 9026*x^12 - 1732*x^10 - 1640*x^8 + 696*x^6 - 31*x^4 - 16*x^2 + 1)) + 36*12^(1/4)*(sqrt(
3)*sqrt(2)*(x^46 + 21*x^44 + 37*x^42 - 36*x^40 + 376*x^38 + 666*x^36 - 3516*x^34 - 4890*x^32 + 10920*x^30 + 13
636*x^28 - 18156*x^26 - 18794*x^24 + 18156*x^22 + 13636*x^20 - 10920*x^18 - 4890*x^16 + 3516*x^14 + 666*x^12 -
376*x^10 - 36*x^8 - 37*x^6 + 21*x^4 - x^2) - 3*sqrt(2)*(x^46 + 15*x^44 - 29*x^42 - 438*x^40 - 338*x^38 + 2664
*x^36 + 3204*x^34 - 7680*x^32 - 10338*x^30 + 13366*x^28 + 17610*x^26 - 15854*x^24 - 17610*x^22 + 13366*x^20 +
10338*x^18 - 7680*x^16 - 3204*x^14 + 2664*x^12 + 338*x^10 - 438*x^8 + 29*x^6 + 15*x^4 - x^2)))*sqrt(sqrt(3) +
2))*sqrt((12*x^6 + 12*x^4 + 12^(1/4)*sqrt(x^4 + x^2 - 1)*(3*sqrt(2)*x^3 + sqrt(3)*sqrt(2)*(x^5 + x^3 - x))*sqr
t(sqrt(3) + 2) - 12*x^2 + sqrt(3)*(x^8 + 5*x^6 + 5*x^4 - 5*x^2 + 1))/(x^8 - x^6 - x^4 + x^2 + 1)) + 72*sqrt(3)
*(3*x^46 + 51*x^44 + 21*x^42 - 1284*x^40 - 2328*x^38 + 8136*x^36 + 18288*x^34 - 23832*x^32 - 60048*x^30 + 4105
2*x^28 + 104802*x^26 - 48246*x^24 - 104802*x^22 + 41052*x^20 + 60048*x^18 - 23832*x^16 - 18288*x^14 + 8136*x^1
2 + 2328*x^10 - 1284*x^8 - 21*x^6 + 51*x^4 - 3*x^2 - sqrt(3)*(x^46 + 25*x^44 + 103*x^42 - 82*x^40 - 908*x^38 -
18*x^36 + 3858*x^34 + 522*x^32 - 9636*x^30 - 1046*x^28 + 15022*x^26 + 1198*x^24 - 15022*x^22 - 1046*x^20 + 96
36*x^18 + 522*x^16 - 3858*x^14 - 18*x^12 + 908*x^10 - 82*x^8 - 103*x^6 + 25*x^4 - x^2)) - 72*sqrt(3)*(x^46 + 3
5*x^44 + 383*x^42 - 14*x^40 - 4664*x^38 - 1596*x^36 + 25464*x^34 + 10866*x^32 - 74448*x^30 - 30740*x^28 + 1252
94*x^26 + 42896*x^24 - 125294*x^22 - 30740*x^20 + 74448*x^18 + 10866*x^16 - 25464*x^14 - 1596*x^12 + 4664*x^10
- 14*x^8 - 383*x^6 + 35*x^4 - x^2) + 36)/(x^48 + 18*x^46 - 111*x^44 - 2552*x^42 - 3606*x^40 + 27594*x^38 + 53
426*x^36 - 113958*x^34 - 252837*x^32 + 250858*x^30 + 592002*x^28 - 353226*x^26 - 777749*x^24 + 353226*x^22 + 5
92002*x^20 - 250858*x^18 - 252837*x^16 + 113958*x^14 + 53426*x^12 - 27594*x^10 - 3606*x^8 + 2552*x^6 - 111*x^4
- 18*x^2 + 1)) + 1/2*log(-(x^4 + 2*x^2 - 2*sqrt(x^4 + x^2 - 1)*x - 1)/(x^4 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 2.83, size = 566, normalized size = 4.42

method	result	size
trager	\(-\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}+\RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right ) \ln \left (-\frac {8 x^{2} \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{5}+2 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{4}+8 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{2}-6 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \sqrt {x^{4}+x^{2}-1}\, x -2 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3}-3 x \sqrt {x^{4}+x^{2}-1}}{4 x^{2} \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}-x^{4}+2 x^{2}+1}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \ln \left (\frac {16 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{4} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{4}+8 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) x^{4}-24 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \sqrt {x^{4}+x^{2}-1}\, x +4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )-3 \RootOf \left (\textit {\_Z}^{2}+4 \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 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )}{4 x^{2} \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+x^{4}+x^{2}-1}\right )}{2}\)	\(566\)
elliptic	\(\frac {\left (-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}\, \sqrt {3}}{4}+\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}+\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}\, \sqrt {3}}{4}-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}-\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )\right ) \sqrt {2}}{2}\)	\(599\)
default	\(-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{6}-\textit {\_Z}^{4}+\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{6}-2 \underline {\hspace {1.25 ex}}\alpha ^{4}+1\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (20 \underline {\hspace {1.25 ex}}\alpha ^{6}-30 \underline {\hspace {1.25 ex}}\alpha ^{4}-5 \underline {\hspace {1.25 ex}}\alpha ^{2}+7 x^{2}+26\right )}{14 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{2}-1}\, \sqrt {x^{4}+x^{2}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{2}-1}}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{7}+\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-x^{2}+2+\sqrt {5}\, x^{2}}\, \sqrt {-x^{2}+2-\sqrt {5}\, x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{6} \sqrt {5}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{6}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {5}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{4}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {5}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\sqrt {5}}{2}+\frac {1}{2}, \frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\sqrt {5}+1}\, \sqrt {x^{4}+x^{2}-1}}\right )\right )}{4}+\frac {92 \sqrt {1-\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {x^{2}}{2}-\frac {\sqrt {5}\, x^{2}}{2}}\, \EllipticF \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )}{15 \sqrt {2-2 \sqrt {5}}\, \sqrt {x^{4}+x^{2}-1}\, \left (\sqrt {5}+1\right )}-\frac {92 \sqrt {1-\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {x^{2}}{2}-\frac {\sqrt {5}\, x^{2}}{2}}\, \EllipticE \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )}{15 \sqrt {2-2 \sqrt {5}}\, \sqrt {x^{4}+x^{2}-1}\, \left (\sqrt {5}+1\right )}-\frac {92 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )\right )}{15 \sqrt {2-2 \sqrt {5}}\, \sqrt {x^{4}+x^{2}-1}\, \left (\sqrt {5}+1\right )}-\frac {\sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}+x^{2}-1}}-\frac {2 \sqrt {1-\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {x^{2}}{2}-\frac {\sqrt {5}\, x^{2}}{2}}\, \EllipticF \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )}{15 \sqrt {2-2 \sqrt {5}}\, \sqrt {x^{4}+x^{2}-1}}-\frac {\sqrt {1-\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {x^{2}}{2}-\frac {\sqrt {5}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , -\frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}+x^{2}-1}}+\frac {32 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )}{15 \sqrt {2-2 \sqrt {5}}\, \sqrt {x^{4}+x^{2}-1}}\)	\(871\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

-1/2*ln(-(x^4+2*x*(x^4+x^2-1)^(1/2)+2*x^2-1)/(-1+x)/(1+x)/(x^2+1))+RootOf(16*_Z^4+12*_Z^2+3)*ln(-(8*x^2*RootOf
(16*_Z^4+12*_Z^2+3)^5+2*RootOf(16*_Z^4+12*_Z^2+3)^3*x^4+8*RootOf(16*_Z^4+12*_Z^2+3)^3*x^2-6*RootOf(16*_Z^4+12*
_Z^2+3)^2*(x^4+x^2-1)^(1/2)*x-2*RootOf(16*_Z^4+12*_Z^2+3)^3-3*x*(x^4+x^2-1)^(1/2))/(4*x^2*RootOf(16*_Z^4+12*_Z
^2+3)^2-x^4+2*x^2+1))-1/2*RootOf(_Z^2+4*RootOf(16*_Z^4+12*_Z^2+3)^2+3)*ln((16*RootOf(_Z^2+4*RootOf(16*_Z^4+12*
_Z^2+3)^2+3)*RootOf(16*_Z^4+12*_Z^2+3)^4*x^2-4*RootOf(_Z^2+4*RootOf(16*_Z^4+12*_Z^2+3)^2+3)*RootOf(16*_Z^4+12*
_Z^2+3)^2*x^4+8*RootOf(_Z^2+4*RootOf(16*_Z^4+12*_Z^2+3)^2+3)*RootOf(16*_Z^4+12*_Z^2+3)^2*x^2-3*RootOf(_Z^2+4*R
ootOf(16*_Z^4+12*_Z^2+3)^2+3)*x^4-24*RootOf(16*_Z^4+12*_Z^2+3)^2*(x^4+x^2-1)^(1/2)*x+4*RootOf(16*_Z^4+12*_Z^2+
3)^2*RootOf(_Z^2+4*RootOf(16*_Z^4+12*_Z^2+3)^2+3)-3*RootOf(_Z^2+4*RootOf(16*_Z^4+12*_Z^2+3)^2+3)*x^2-6*x*(x^4+
x^2-1)^(1/2)+3*RootOf(_Z^2+4*RootOf(16*_Z^4+12*_Z^2+3)^2+3))/(4*x^2*RootOf(16*_Z^4+12*_Z^2+3)^2+x^4+x^2-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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