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

Optimal. Leaf size=163 \[ -\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x \sqrt {-x^4+2 x^2+1}}{x^4-2 x^2-1}\right )-\tanh ^{-1}\left (\frac {x \sqrt {-x^4+2 x^2+1}}{x^4-2 x^2-1}\right )+\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x \sqrt {-x^4+2 x^2+1}}{x^4-2 x^2-1}\right ) \]

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Rubi [F]  time = 3.44, 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 {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*EllipticE[ArcSin[Sqrt[-1 + Sqrt[2]]*x], -3 - 2*Sqrt[2]])/Sqrt[1 + Sqrt[2]] - ((1 + 2*Sqrt[2] - Sqrt[5])*Ell
ipticF[ArcSin[Sqrt[-1 + Sqrt[2]]*x], -3 - 2*Sqrt[2]])/(2*Sqrt[-1 + Sqrt[2]]) - ((1 + 2*Sqrt[2] + Sqrt[5])*Elli
pticF[ArcSin[Sqrt[-1 + Sqrt[2]]*x], -3 - 2*Sqrt[2]])/(2*Sqrt[-1 + Sqrt[2]]) + EllipticPi[(2*(1 + Sqrt[2]))/(1
- Sqrt[5]), ArcSin[Sqrt[-1 + Sqrt[2]]*x], -3 - 2*Sqrt[2]]/Sqrt[-1 + Sqrt[2]] + EllipticPi[(2*(1 + Sqrt[2]))/(1
+ Sqrt[5]), ArcSin[Sqrt[-1 + Sqrt[2]]*x], -3 - 2*Sqrt[2]]/Sqrt[-1 + Sqrt[2]] + 2*Defer[Int][Sqrt[1 + 2*x^2 -
x^4]/(1 + 3*x^2 - x^4 - 3*x^6 + x^8), x] - Defer[Int][(x^2*Sqrt[1 + 2*x^2 - x^4])/(1 + 3*x^2 - x^4 - 3*x^6 + x
^8), x] - 4*Defer[Int][(x^4*Sqrt[1 + 2*x^2 - x^4])/(1 + 3*x^2 - x^4 - 3*x^6 + x^8), x] + 2*Defer[Int][(x^6*Sqr
t[1 + 2*x^2 - x^4])/(1 + 3*x^2 - x^4 - 3*x^6 + x^8), x]

Rubi steps

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

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Mathematica [C]  time = 7.79, size = 11852, normalized size = 72.71 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 1.10, size = 163, normalized size = 1.00 \begin {gather*} -\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x \sqrt {1+2 x^2-x^4}}{-1-2 x^2+x^4}\right )-\tanh ^{-1}\left (\frac {x \sqrt {1+2 x^2-x^4}}{-1-2 x^2+x^4}\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x \sqrt {1+2 x^2-x^4}}{-1-2 x^2+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x*Sqrt[1 + 2*x^2 - x^4])/(-1 - 2*x^2 + x^4)]) - ArcTan
h[(x*Sqrt[1 + 2*x^2 - x^4])/(-1 - 2*x^2 + x^4)] + Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x*Sqr
t[1 + 2*x^2 - x^4])/(-1 - 2*x^2 + x^4)]

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fricas [B]  time = 1.53, size = 485, normalized size = 2.98 \begin {gather*} \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \arctan \left (-\frac {2 \, \sqrt {10} {\left (5 \, x^{5} - 10 \, x^{3} - \sqrt {5} {\left (x^{5} - x\right )} - 5 \, x\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1} \sqrt {\sqrt {5} + 1} - \sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 25 \, x^{4} + 25 \, x^{2} + \sqrt {5} {\left (x^{8} - 9 \, x^{6} + 13 \, x^{4} + 9 \, x^{2} + 1\right )} + 5\right )} \sqrt {\sqrt {5} + 1} \sqrt {\sqrt {5} - 2}}{20 \, {\left (x^{8} - 5 \, x^{6} + 3 \, x^{4} + 5 \, x^{2} + 1\right )}}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\frac {\sqrt {10} {\left (5 \, x^{8} - 35 \, x^{6} + 45 \, x^{4} + 35 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 17 \, x^{6} + 19 \, x^{4} + 17 \, x^{2} + 3\right )} + 5\right )} \sqrt {\sqrt {5} - 1} + 20 \, {\left (3 \, x^{5} - 7 \, x^{3} + \sqrt {5} {\left (x^{5} - 3 \, x^{3} - x\right )} - 3 \, x\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1}}{x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (-\frac {\sqrt {10} {\left (5 \, x^{8} - 35 \, x^{6} + 45 \, x^{4} + 35 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 17 \, x^{6} + 19 \, x^{4} + 17 \, x^{2} + 3\right )} + 5\right )} \sqrt {\sqrt {5} - 1} - 20 \, {\left (3 \, x^{5} - 7 \, x^{3} + \sqrt {5} {\left (x^{5} - 3 \, x^{3} - x\right )} - 3 \, x\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1}}{x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - 3 \, x^{2} - 2 \, \sqrt {-x^{4} + 2 \, x^{2} + 1} x - 1}{x^{4} - x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(10)*sqrt(sqrt(5) + 1)*arctan(-1/20*(2*sqrt(10)*(5*x^5 - 10*x^3 - sqrt(5)*(x^5 - x) - 5*x)*sqrt(-x^4
+ 2*x^2 + 1)*sqrt(sqrt(5) + 1) - sqrt(10)*(5*x^8 - 25*x^6 + 25*x^4 + 25*x^2 + sqrt(5)*(x^8 - 9*x^6 + 13*x^4 +
9*x^2 + 1) + 5)*sqrt(sqrt(5) + 1)*sqrt(sqrt(5) - 2))/(x^8 - 5*x^6 + 3*x^4 + 5*x^2 + 1)) + 1/40*sqrt(10)*sqrt(s
qrt(5) - 1)*log((sqrt(10)*(5*x^8 - 35*x^6 + 45*x^4 + 35*x^2 + sqrt(5)*(3*x^8 - 17*x^6 + 19*x^4 + 17*x^2 + 3) +
5)*sqrt(sqrt(5) - 1) + 20*(3*x^5 - 7*x^3 + sqrt(5)*(x^5 - 3*x^3 - x) - 3*x)*sqrt(-x^4 + 2*x^2 + 1))/(x^8 - 3*
x^6 - x^4 + 3*x^2 + 1)) - 1/40*sqrt(10)*sqrt(sqrt(5) - 1)*log(-(sqrt(10)*(5*x^8 - 35*x^6 + 45*x^4 + 35*x^2 + s
qrt(5)*(3*x^8 - 17*x^6 + 19*x^4 + 17*x^2 + 3) + 5)*sqrt(sqrt(5) - 1) - 20*(3*x^5 - 7*x^3 + sqrt(5)*(x^5 - 3*x^
3 - x) - 3*x)*sqrt(-x^4 + 2*x^2 + 1))/(x^8 - 3*x^6 - x^4 + 3*x^2 + 1)) + 1/2*log(-(x^4 - 3*x^2 - 2*sqrt(-x^4 +
2*x^2 + 1)*x - 1)/(x^4 - x^2 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )} {\left (x^{4} - 1\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1}}{{\left (x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 3.30, size = 112, normalized size = 0.69

method	result	size
elliptic	\(\frac {\left (-\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}-\frac {2 \sqrt {5}\, \arctanh \left (\frac {\sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\sqrt {2}\, \arctanh \left (\frac {\sqrt {-x^{4}+2 x^{2}+1}}{x}\right )\right ) \sqrt {2}}{2}\)	\(112\)
default	\(-\frac {\sqrt {1-\left (\sqrt {2}-1\right ) x^{2}}\, \sqrt {1-\left (-1-\sqrt {2}\right ) x^{2}}\, \EllipticF \left (x \sqrt {\sqrt {2}-1}, i+i \sqrt {2}\right )}{\sqrt {\sqrt {2}-1}\, \sqrt {-x^{4}+2 x^{2}+1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}-3 \textit {\_Z}^{6}-\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{6}-6 \underline {\hspace {1.25 ex}}\alpha ^{4}+3\right ) \left (\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \left (2 \underline {\hspace {1.25 ex}}\alpha ^{6}-4 \underline {\hspace {1.25 ex}}\alpha ^{4}-6 \underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{4}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1}\, \sqrt {-x^{4}+2 x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{4}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1}}-\frac {2 \left (-\underline {\hspace {1.25 ex}}\alpha ^{7}+3 \underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\sqrt {2}\, x^{2}+x^{2}+1}\, \sqrt {\sqrt {2}\, x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {\sqrt {2}-1}, -\underline {\hspace {1.25 ex}}\alpha ^{6} \sqrt {2}-\underline {\hspace {1.25 ex}}\alpha ^{6}+3 \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {2}+3 \underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}+\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \sqrt {2}-3, \frac {\sqrt {-1-\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {\sqrt {2}-1}\, \sqrt {-x^{4}+2 x^{2}+1}}\right )\right )}{20}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {-x^{4}+2 x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\sqrt {2}\, x^{2}+x^{2}+1}\, \sqrt {\sqrt {2}\, x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {\sqrt {2}-1}, \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}+\underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {2}-1, \frac {\sqrt {-1-\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {\sqrt {2}-1}\, \sqrt {-x^{4}+2 x^{2}+1}}\right )\right )}{4}\)	\(488\)
trager	\(\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {-x^{4}+2 x^{2}+1}\, x +3 x^{2}+1}{x^{4}-x^{2}-1}\right )}{2}-\RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {600 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{5}-30 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{3} x^{4}+130 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{3} x^{2}-2 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right ) x^{4}+10 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} \sqrt {-x^{4}+2 x^{2}+1}\, x +30 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{3}+6 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right ) x^{2}+\sqrt {-x^{4}+2 x^{2}+1}\, x +2 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+x^{4}-x^{2}-1}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \ln \left (-\frac {1200 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{4} x^{2}+60 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} x^{4}-140 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{4}+200 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} \sqrt {-x^{4}+2 x^{2}+1}\, x -60 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right )+2 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{2}-10 \sqrt {-x^{4}+2 x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}-x^{4}+2 x^{2}+1}\right )}{10}\)	\(634\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-2/5*5^(1/2)/(5^(1/2)-1)^(1/2)*arctan((-x^4+2*x^2+1)^(1/2)*2^(1/2)/x/(5^(1/2)-1)^(1/2))-2/5*5^(1/2)/(5^(1
/2)+1)^(1/2)*arctanh((-x^4+2*x^2+1)^(1/2)*2^(1/2)/x/(5^(1/2)+1)^(1/2))+2^(1/2)*arctanh((-x^4+2*x^2+1)^(1/2)/x)
)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )} {\left (x^{4} - 1\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1}}{{\left (x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + 1)*(x^4 - 1)*sqrt(-x^4 + 2*x^2 + 1)/((x^8 - 3*x^6 - x^4 + 3*x^2 + 1)*(x^4 - x^2 - 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+1\right )\,\sqrt {-x^4+2\,x^2+1}}{\left (-x^4+x^2+1\right )\,\left (x^8-3\,x^6-x^4+3\,x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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