3.19.20 \(\int \frac {(1+x^6)^2 (-1+2 x^6)}{(1-x^2+x^6)^{3/2} (1-x^2-x^4+2 x^6-x^8+x^{12})} \, dx\)

Optimal. Leaf size=123 \[ \frac {x}{\sqrt {x^6-x^2+1}}-\sqrt {\frac {2}{145+65 \sqrt {5}}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {x^6-x^2+1}}\right )-\sqrt {\frac {1}{10} \left (29+13 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt {x^6-x^2+1}}\right ) \]

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Rubi [F]  time = 2.97, 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^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)),x]

[Out]

-(x/Sqrt[1 - x^2 + x^6]) + 2*Defer[Int][x^2/(1 - x^2 + x^6)^(3/2), x] - 3*Defer[Int][x^2/((1 - x^2 + x^6)^(3/2
)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)), x] + Defer[Int][x^4/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x
^8 + x^12)), x] + 2*Defer[Int][x^6/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)), x] - 3*Defer[
Int][x^8/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)), x] + 4*Defer[Int][x^10/((1 - x^2 + x^6)
^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)), x]

Rubi steps

\begin {align*} \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx &=\int \left (-\frac {1}{\left (1-x^2+x^6\right )^{3/2}}+\frac {2 x^2}{\left (1-x^2+x^6\right )^{3/2}}+\frac {2 x^6}{\left (1-x^2+x^6\right )^{3/2}}+\frac {x^2 \left (-3+x^2+2 x^4-3 x^6+4 x^8\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )}\right ) \, dx\\ &=2 \int \frac {x^2}{\left (1-x^2+x^6\right )^{3/2}} \, dx+2 \int \frac {x^6}{\left (1-x^2+x^6\right )^{3/2}} \, dx-\int \frac {1}{\left (1-x^2+x^6\right )^{3/2}} \, dx+\int \frac {x^2 \left (-3+x^2+2 x^4-3 x^6+4 x^8\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx\\ &=-\frac {x}{\sqrt {1-x^2+x^6}}+2 \int \frac {x^2}{\left (1-x^2+x^6\right )^{3/2}} \, dx+\int \left (-\frac {3 x^2}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )}+\frac {x^4}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )}+\frac {2 x^6}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )}-\frac {3 x^8}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )}+\frac {4 x^{10}}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )}\right ) \, dx\\ &=-\frac {x}{\sqrt {1-x^2+x^6}}+2 \int \frac {x^2}{\left (1-x^2+x^6\right )^{3/2}} \, dx+2 \int \frac {x^6}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx-3 \int \frac {x^2}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx-3 \int \frac {x^8}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx+4 \int \frac {x^{10}}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx+\int \frac {x^4}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)),x]

[Out]

Integrate[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)), x]

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IntegrateAlgebraic [A]  time = 3.34, size = 123, normalized size = 1.00 \begin {gather*} \frac {x}{\sqrt {1-x^2+x^6}}-\sqrt {\frac {2}{145+65 \sqrt {5}}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1-x^2+x^6}}\right )-\sqrt {\frac {1}{10} \left (29+13 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1-x^2+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)),x]

[Out]

x/Sqrt[1 - x^2 + x^6] - Sqrt[2/(145 + 65*Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/Sqrt[1 - x^2 + x^6]] - Sqr
t[(29 + 13*Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/Sqrt[1 - x^2 + x^6]]

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fricas [B]  time = 1.02, size = 555, normalized size = 4.51 \begin {gather*} -\frac {4 \, \sqrt {10} {\left (x^{6} - x^{2} + 1\right )} \sqrt {13 \, \sqrt {5} - 29} \arctan \left (\frac {2 \, \sqrt {10} {\left (15 \, x^{7} - 5 \, x^{3} + \sqrt {5} {\left (7 \, x^{7} - 3 \, x^{3} + 7 \, x\right )} + 15 \, x\right )} \sqrt {x^{6} - x^{2} + 1} \sqrt {13 \, \sqrt {5} - 29} + \sqrt {10} {\left (5 \, x^{12} + 5 \, x^{8} + 10 \, x^{6} - 5 \, x^{4} + 5 \, x^{2} + \sqrt {5} {\left (3 \, x^{12} - x^{8} + 6 \, x^{6} + x^{4} - x^{2} + 3\right )} + 5\right )} \sqrt {13 \, \sqrt {5} - 29} \sqrt {\sqrt {5} + 2}}{20 \, {\left (x^{12} - 3 \, x^{8} + 2 \, x^{6} + x^{4} - 3 \, x^{2} + 1\right )}}\right ) + \sqrt {10} {\left (x^{6} - x^{2} + 1\right )} \sqrt {13 \, \sqrt {5} + 29} \log \left (-\frac {\sqrt {10} {\left (25 \, x^{12} - 105 \, x^{8} + 50 \, x^{6} + 105 \, x^{4} - 105 \, x^{2} - \sqrt {5} {\left (11 \, x^{12} - 47 \, x^{8} + 22 \, x^{6} + 47 \, x^{4} - 47 \, x^{2} + 11\right )} + 25\right )} \sqrt {13 \, \sqrt {5} + 29} + 20 \, {\left (3 \, x^{7} - 4 \, x^{3} - \sqrt {5} {\left (x^{7} - 2 \, x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + 1}}{x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1}\right ) - \sqrt {10} {\left (x^{6} - x^{2} + 1\right )} \sqrt {13 \, \sqrt {5} + 29} \log \left (\frac {\sqrt {10} {\left (25 \, x^{12} - 105 \, x^{8} + 50 \, x^{6} + 105 \, x^{4} - 105 \, x^{2} - \sqrt {5} {\left (11 \, x^{12} - 47 \, x^{8} + 22 \, x^{6} + 47 \, x^{4} - 47 \, x^{2} + 11\right )} + 25\right )} \sqrt {13 \, \sqrt {5} + 29} - 20 \, {\left (3 \, x^{7} - 4 \, x^{3} - \sqrt {5} {\left (x^{7} - 2 \, x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + 1}}{x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1}\right ) - 40 \, \sqrt {x^{6} - x^{2} + 1} x}{40 \, {\left (x^{6} - x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1),x, algorithm="fricas")

[Out]

-1/40*(4*sqrt(10)*(x^6 - x^2 + 1)*sqrt(13*sqrt(5) - 29)*arctan(1/20*(2*sqrt(10)*(15*x^7 - 5*x^3 + sqrt(5)*(7*x
^7 - 3*x^3 + 7*x) + 15*x)*sqrt(x^6 - x^2 + 1)*sqrt(13*sqrt(5) - 29) + sqrt(10)*(5*x^12 + 5*x^8 + 10*x^6 - 5*x^
4 + 5*x^2 + sqrt(5)*(3*x^12 - x^8 + 6*x^6 + x^4 - x^2 + 3) + 5)*sqrt(13*sqrt(5) - 29)*sqrt(sqrt(5) + 2))/(x^12
- 3*x^8 + 2*x^6 + x^4 - 3*x^2 + 1)) + sqrt(10)*(x^6 - x^2 + 1)*sqrt(13*sqrt(5) + 29)*log(-(sqrt(10)*(25*x^12
- 105*x^8 + 50*x^6 + 105*x^4 - 105*x^2 - sqrt(5)*(11*x^12 - 47*x^8 + 22*x^6 + 47*x^4 - 47*x^2 + 11) + 25)*sqrt
(13*sqrt(5) + 29) + 20*(3*x^7 - 4*x^3 - sqrt(5)*(x^7 - 2*x^3 + x) + 3*x)*sqrt(x^6 - x^2 + 1))/(x^12 - x^8 + 2*
x^6 - x^4 - x^2 + 1)) - sqrt(10)*(x^6 - x^2 + 1)*sqrt(13*sqrt(5) + 29)*log((sqrt(10)*(25*x^12 - 105*x^8 + 50*x
^6 + 105*x^4 - 105*x^2 - sqrt(5)*(11*x^12 - 47*x^8 + 22*x^6 + 47*x^4 - 47*x^2 + 11) + 25)*sqrt(13*sqrt(5) + 29
) - 20*(3*x^7 - 4*x^3 - sqrt(5)*(x^7 - 2*x^3 + x) + 3*x)*sqrt(x^6 - x^2 + 1))/(x^12 - x^8 + 2*x^6 - x^4 - x^2
+ 1)) - 40*sqrt(x^6 - x^2 + 1)*x)/(x^6 - x^2 + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}^{2}}{{\left (x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1\right )} {\left (x^{6} - x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1),x, algorithm="giac")

[Out]

integrate((2*x^6 - 1)*(x^6 + 1)^2/((x^12 - x^8 + 2*x^6 - x^4 - x^2 + 1)*(x^6 - x^2 + 1)^(3/2)), x)

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maple [C]  time = 7.80, size = 599, normalized size = 4.87

method	result	size
trager	\(\frac {x}{\sqrt {x^{6}-x^{2}+1}}-\RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {1430 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{6}+2200 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{5} x^{2}-2093 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{6}-6960 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{2}+130 \sqrt {x^{6}-x^{2}+1}\, \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x +1430 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3}+5474 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{2}-221 \sqrt {x^{6}-x^{2}+1}\, x -2093 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )}{-13 x^{6}+20 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-8 x^{2}-13}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \ln \left (-\frac {-2860 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{6}+4400 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{4} x^{2}-39 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{6}+1160 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}+2600 \sqrt {x^{6}-x^{2}+1}\, \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x -2860 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )+15 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{2}+650 \sqrt {x^{6}-x^{2}+1}\, x -39 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )}{13 x^{6}+20 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-21 x^{2}+13}\right )}{10}\)	\(599\)
risch	\(\frac {x}{\sqrt {x^{6}-x^{2}+1}}-\RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {1430 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{6}+2200 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{5} x^{2}-2093 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{6}-6960 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{2}+130 \sqrt {x^{6}-x^{2}+1}\, \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x +1430 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3}+5474 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{2}-221 \sqrt {x^{6}-x^{2}+1}\, x -2093 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )}{-13 x^{6}+20 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-8 x^{2}-13}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \ln \left (-\frac {2860 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{6}-4400 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{4} x^{2}+39 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{6}-1160 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}+2600 \sqrt {x^{6}-x^{2}+1}\, \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x +2860 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )-15 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{2}+650 \sqrt {x^{6}-x^{2}+1}\, x +39 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )}{13 x^{6}+20 \RootOf \left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-21 x^{2}+13}\right )}{10}\)	\(599\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1),x,method=_RETURNVERBOSE)

[Out]

x/(x^6-x^2+1)^(1/2)-RootOf(400*_Z^4-580*_Z^2-1)*ln((1430*RootOf(400*_Z^4-580*_Z^2-1)^3*x^6+2200*RootOf(400*_Z^
4-580*_Z^2-1)^5*x^2-2093*RootOf(400*_Z^4-580*_Z^2-1)*x^6-6960*RootOf(400*_Z^4-580*_Z^2-1)^3*x^2+130*(x^6-x^2+1
)^(1/2)*RootOf(400*_Z^4-580*_Z^2-1)^2*x+1430*RootOf(400*_Z^4-580*_Z^2-1)^3+5474*RootOf(400*_Z^4-580*_Z^2-1)*x^
2-221*(x^6-x^2+1)^(1/2)*x-2093*RootOf(400*_Z^4-580*_Z^2-1))/(-13*x^6+20*RootOf(400*_Z^4-580*_Z^2-1)^2*x^2-8*x^
2-13))+1/10*RootOf(_Z^2+100*RootOf(400*_Z^4-580*_Z^2-1)^2-145)*ln(-(-2860*RootOf(_Z^2+100*RootOf(400*_Z^4-580*
_Z^2-1)^2-145)*RootOf(400*_Z^4-580*_Z^2-1)^2*x^6+4400*RootOf(_Z^2+100*RootOf(400*_Z^4-580*_Z^2-1)^2-145)*RootO
f(400*_Z^4-580*_Z^2-1)^4*x^2-39*RootOf(_Z^2+100*RootOf(400*_Z^4-580*_Z^2-1)^2-145)*x^6+1160*RootOf(_Z^2+100*Ro
otOf(400*_Z^4-580*_Z^2-1)^2-145)*RootOf(400*_Z^4-580*_Z^2-1)^2*x^2+2600*(x^6-x^2+1)^(1/2)*RootOf(400*_Z^4-580*
_Z^2-1)^2*x-2860*RootOf(400*_Z^4-580*_Z^2-1)^2*RootOf(_Z^2+100*RootOf(400*_Z^4-580*_Z^2-1)^2-145)+15*RootOf(_Z
^2+100*RootOf(400*_Z^4-580*_Z^2-1)^2-145)*x^2+650*(x^6-x^2+1)^(1/2)*x-39*RootOf(_Z^2+100*RootOf(400*_Z^4-580*_
Z^2-1)^2-145))/(13*x^6+20*RootOf(400*_Z^4-580*_Z^2-1)^2*x^2-21*x^2+13))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}^{2}}{{\left (x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1\right )} {\left (x^{6} - x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1),x, algorithm="maxima")

[Out]

integrate((2*x^6 - 1)*(x^6 + 1)^2/((x^12 - x^8 + 2*x^6 - x^4 - x^2 + 1)*(x^6 - x^2 + 1)^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (x^6+1\right )}^2\,\left (2\,x^6-1\right )}{{\left (x^6-x^2+1\right )}^{3/2}\,\left (-x^{12}+x^8-2\,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^6 + 1)^2*(2*x^6 - 1))/((x^6 - x^2 + 1)^(3/2)*(x^2 + x^4 - 2*x^6 + x^8 - x^12 - 1)),x)

[Out]

int(-((x^6 + 1)^2*(2*x^6 - 1))/((x^6 - x^2 + 1)^(3/2)*(x^2 + x^4 - 2*x^6 + x^8 - x^12 - 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**6+1)**2*(2*x**6-1)/(x**6-x**2+1)**(3/2)/(x**12-x**8+2*x**6-x**4-x**2+1),x)

[Out]

Timed out

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