3.15.96 \(\int \frac {(-2+x^6) (4+x^6) \sqrt [4]{-2+2 x^4+x^6}}{x^6 (-4-x^4+2 x^6)} \, dx\)

Optimal. Leaf size=104 \[ \frac {1}{4} \sqrt [4]{\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{x^6+2 x^4-2}}\right )-\frac {1}{4} \sqrt [4]{\frac {5}{2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{x^6+2 x^4-2}}\right )+\frac {\sqrt [4]{x^6+2 x^4-2} \left (2 x^6+9 x^4-4\right )}{10 x^5} \]

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Rubi [F]  time = 1.73, 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 (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.80, size = 104, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-2+2 x^4+x^6} \left (-4+9 x^4+2 x^6\right )}{10 x^5}+\frac {1}{4} \sqrt [4]{\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right )-\frac {1}{4} \sqrt [4]{\frac {5}{2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 + 2*x^4 + x^6)^(1/4)*(-4 + 9*x^4 + 2*x^6))/(10*x^5) + ((5/2)^(1/4)*ArcTan[((5/2)^(1/4)*x)/(-2 + 2*x^4 + x
^6)^(1/4)])/4 - ((5/2)^(1/4)*ArcTanh[((5/2)^(1/4)*x)/(-2 + 2*x^4 + x^6)^(1/4)])/4

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fricas [B]  time = 142.92, size = 380, normalized size = 3.65 \begin {gather*} \frac {20 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \arctan \left (\frac {20 \cdot 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 20 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {5} {\left (4 \cdot 5^{\frac {3}{4}} 2^{\frac {1}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} + 5^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )}\right )} \sqrt {\sqrt {5} \sqrt {2}}}{10 \, {\left (2 \, x^{6} - x^{4} - 4\right )}}\right ) - 5 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 10 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} + 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} + 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) + 5 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - 10 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} - 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} + 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) + 16 \, {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}}}{160 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/160*(20*5^(1/4)*2^(3/4)*x^5*arctan(1/10*(20*5^(3/4)*2^(1/4)*(x^6 + 2*x^4 - 2)^(1/4)*x^3 + 20*5^(1/4)*2^(3/4)
*(x^6 + 2*x^4 - 2)^(3/4)*x + sqrt(5)*(4*5^(3/4)*2^(1/4)*sqrt(x^6 + 2*x^4 - 2)*x^2 + 5^(1/4)*2^(3/4)*(2*x^6 + 9
*x^4 - 4))*sqrt(sqrt(5)*sqrt(2)))/(2*x^6 - x^4 - 4)) - 5*5^(1/4)*2^(3/4)*x^5*log(-(10*sqrt(5)*sqrt(2)*(x^6 + 2
*x^4 - 2)^(1/4)*x^3 + 10*5^(1/4)*2^(3/4)*sqrt(x^6 + 2*x^4 - 2)*x^2 + 5^(3/4)*2^(1/4)*(2*x^6 + 9*x^4 - 4) + 20*
(x^6 + 2*x^4 - 2)^(3/4)*x)/(2*x^6 - x^4 - 4)) + 5*5^(1/4)*2^(3/4)*x^5*log(-(10*sqrt(5)*sqrt(2)*(x^6 + 2*x^4 -
2)^(1/4)*x^3 - 10*5^(1/4)*2^(3/4)*sqrt(x^6 + 2*x^4 - 2)*x^2 - 5^(3/4)*2^(1/4)*(2*x^6 + 9*x^4 - 4) + 20*(x^6 +
2*x^4 - 2)^(3/4)*x)/(2*x^6 - x^4 - 4)) + 16*(2*x^6 + 9*x^4 - 4)*(x^6 + 2*x^4 - 2)^(1/4))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 61.58, size = 333, normalized size = 3.20

method	result	size
trager	\(\frac {\left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} \left (2 x^{6}+9 x^{4}-4\right )}{10 x^{5}}+\frac {\RootOf \left (\textit {\_Z}^{4}-40\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x^{6}+9 \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x^{4}-20 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} x^{3}+40 \RootOf \left (\textit {\_Z}^{4}-40\right ) \sqrt {x^{6}+2 x^{4}-2}\, x^{2}-80 \left (x^{6}+2 x^{4}-2\right )^{\frac {3}{4}} x -4 \RootOf \left (\textit {\_Z}^{4}-40\right )^{3}}{2 x^{6}-x^{4}-4}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) x^{6}+9 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) x^{4}-20 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} x^{3}-40 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \sqrt {x^{6}+2 x^{4}-2}\, x^{2}+80 \left (x^{6}+2 x^{4}-2\right )^{\frac {3}{4}} x -4 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right )}{2 x^{6}-x^{4}-4}\right )}{16}\)	\(333\)
risch	\(\frac {2 x^{12}+13 x^{10}+18 x^{8}-8 x^{6}-26 x^{4}+8}{10 x^{5} \left (x^{6}+2 x^{4}-2\right )^{\frac {3}{4}}}+\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{4}-40\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{18}+17 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{16}+2 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x^{13}+44 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{14}+8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x^{11}+24 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{12}+8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x^{9}-68 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{10}-8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x^{7}-88 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{8}+40 \sqrt {x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8}\, x^{8}-16 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x^{5}+24 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{6}+80 \sqrt {x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8}\, x^{6}+20 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}-40\right ) x^{3}+68 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{4}+8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-40\right )^{3} x -80 \sqrt {x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8}\, x^{2}-16 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2}}{\left (2 x^{6}-x^{4}-4\right ) \left (x^{6}+2 x^{4}-2\right )^{2}}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{18}-17 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{16}+2 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{13}-44 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{14}+8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{11}-24 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{12}+8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{9}+68 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{10}-8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{7}+88 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{8}+40 \sqrt {x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8}\, x^{8}-16 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{5}-24 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{6}+80 \sqrt {x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8}\, x^{6}-20 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {3}{4}} x^{3}-68 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x^{4}+8 \left (x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-40\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-40\right )^{2} x -80 \sqrt {x^{18}+6 x^{16}+12 x^{14}+2 x^{12}-24 x^{10}-24 x^{8}+12 x^{6}+24 x^{4}-8}\, x^{2}+16 \RootOf \left (\textit {\_Z}^{4}-40\right )^{2}}{\left (2 x^{6}-x^{4}-4\right ) \left (x^{6}+2 x^{4}-2\right )^{2}}\right )}{16}\right ) \left (\left (x^{6}+2 x^{4}-2\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{6}+2 x^{4}-2\right )^{\frac {3}{4}}}\)	\(1514\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(x^6+2*x^4-2)^(1/4)*(2*x^6+9*x^4-4)/x^5+1/16*RootOf(_Z^4-40)*ln((2*RootOf(_Z^4-40)^3*x^6+9*RootOf(_Z^4-40
)^3*x^4-20*RootOf(_Z^4-40)^2*(x^6+2*x^4-2)^(1/4)*x^3+40*RootOf(_Z^4-40)*(x^6+2*x^4-2)^(1/2)*x^2-80*(x^6+2*x^4-
2)^(3/4)*x-4*RootOf(_Z^4-40)^3)/(2*x^6-x^4-4))+1/16*RootOf(_Z^2+RootOf(_Z^4-40)^2)*ln(-(2*RootOf(_Z^4-40)^2*Ro
otOf(_Z^2+RootOf(_Z^4-40)^2)*x^6+9*RootOf(_Z^4-40)^2*RootOf(_Z^2+RootOf(_Z^4-40)^2)*x^4-20*RootOf(_Z^4-40)^2*(
x^6+2*x^4-2)^(1/4)*x^3-40*RootOf(_Z^2+RootOf(_Z^4-40)^2)*(x^6+2*x^4-2)^(1/2)*x^2+80*(x^6+2*x^4-2)^(3/4)*x-4*Ro
otOf(_Z^4-40)^2*RootOf(_Z^2+RootOf(_Z^4-40)^2))/(2*x^6-x^4-4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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