3.23.79 \(\int \frac {(1+2 x^8) \sqrt [4]{-1-2 x^4+2 x^8} (1-3 x^8+4 x^{16})}{x^{10} (-1+2 x^8)} \, dx\) [2279]

Optimal. Leaf size=173 \[ \frac {\sqrt [4]{-1-2 x^4+2 x^8} \left (5+2 x^4+9 x^8-4 x^{12}+20 x^{16}\right )}{45 x^9}-\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt {2} x^2-\sqrt {-1-2 x^4+2 x^8}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-2 x^4+2 x^8}}{2 x^2+\sqrt {2} \sqrt {-1-2 x^4+2 x^8}}\right )}{2 \sqrt [4]{2}} \]

[Out]

1/45*(2*x^8-2*x^4-1)^(1/4)*(20*x^16-4*x^12+9*x^8+2*x^4+5)/x^9-1/4*arctan(2^(3/4)*x*(2*x^8-2*x^4-1)^(1/4)/(x^2*
2^(1/2)-(2*x^8-2*x^4-1)^(1/2)))*2^(3/4)-1/4*arctanh(2*2^(1/4)*x*(2*x^8-2*x^4-1)^(1/4)/(2*x^2+2^(1/2)*(2*x^8-2*
x^4-1)^(1/2)))*2^(3/4)

________________________________________________________________________________________

Rubi [F]
time = 2.03, 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+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + 2*x^8)*(-1 - 2*x^4 + 2*x^8)^(1/4)*(1 - 3*x^8 + 4*x^16))/(x^10*(-1 + 2*x^8)),x]

[Out]

((-1 - 2*x^4 + 2*x^8)^(1/4)*AppellF1[-9/4, -1/4, -1/4, -5/4, (2*x^4)/(1 + Sqrt[3]), (2*x^4)/(1 - Sqrt[3])])/(9
*x^9*(1 - (2*x^4)/(1 - Sqrt[3]))^(1/4)*(1 - (2*x^4)/(1 + Sqrt[3]))^(1/4)) + ((-1 - 2*x^4 + 2*x^8)^(1/4)*Appell
F1[-1/4, -1/4, -1/4, 3/4, (2*x^4)/(1 + Sqrt[3]), (2*x^4)/(1 - Sqrt[3])])/(x*(1 - (2*x^4)/(1 - Sqrt[3]))^(1/4)*
(1 - (2*x^4)/(1 + Sqrt[3]))^(1/4)) + (4*x^7*(-1 - 2*x^4 + 2*x^8)^(1/4)*AppellF1[7/4, -1/4, -1/4, 11/4, (2*x^4)
/(1 + Sqrt[3]), (2*x^4)/(1 - Sqrt[3])])/(7*(1 - (2*x^4)/(1 - Sqrt[3]))^(1/4)*(1 - (2*x^4)/(1 + Sqrt[3]))^(1/4)
) + ((I/2)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(I - 2^(1/8)*x), x])/2^(3/4) - Defer[Int][(-1 - 2*x^4 + 2*x^8
)^(1/4)/(1 - 2^(1/8)*x), x]/(2*2^(3/4)) - ((1/4 - I/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/((-1)^(1/4) - 2^(
1/8)*x), x])/2^(1/4) - ((-1)^(1/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(-(-1)^(3/4) - 2^(1/8)*x), x])/(2*2^(
3/4)) + ((I/2)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(I + 2^(1/8)*x), x])/2^(3/4) - Defer[Int][(-1 - 2*x^4 + 2
*x^8)^(1/4)/(1 + 2^(1/8)*x), x]/(2*2^(3/4)) - ((1/4 - I/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/((-1)^(1/4) +
2^(1/8)*x), x])/2^(1/4) - ((-1)^(1/4)*Defer[Int][(-1 - 2*x^4 + 2*x^8)^(1/4)/(-(-1)^(3/4) + 2^(1/8)*x), x])/(2
*2^(3/4))

Rubi steps

\begin {align*} \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx &=\int \left (-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^{10}}-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^2}+4 x^6 \sqrt [4]{-1-2 x^4+2 x^8}+\frac {4 x^6 \sqrt [4]{-1-2 x^4+2 x^8}}{-1+2 x^8}\right ) \, dx\\ &=4 \int x^6 \sqrt [4]{-1-2 x^4+2 x^8} \, dx+4 \int \frac {x^6 \sqrt [4]{-1-2 x^4+2 x^8}}{-1+2 x^8} \, dx-\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^{10}} \, dx-\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{x^2} \, dx\\ &=4 \int \left (\frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (-\sqrt {2}+2 x^4\right )}+\frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (\sqrt {2}+2 x^4\right )}\right ) \, dx-\frac {\sqrt [4]{-1-2 x^4+2 x^8} \int \frac {\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}{x^{10}} \, dx}{\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}-\frac {\sqrt [4]{-1-2 x^4+2 x^8} \int \frac {\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}+\frac {\left (4 \sqrt [4]{-1-2 x^4+2 x^8}\right ) \int x^6 \sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}} \, dx}{\sqrt [4]{1+\frac {4 x^4}{-2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^4}{-2+2 \sqrt {3}}}}\\ &=\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {9}{4};-\frac {1}{4},-\frac {1}{4};-\frac {5}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {1}{4};-\frac {1}{4},-\frac {1}{4};\frac {3}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} F_1\left (\frac {7}{4};-\frac {1}{4},-\frac {1}{4};\frac {11}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+2 \int \frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{-\sqrt {2}+2 x^4} \, dx+2 \int \frac {x^2 \sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt {2}+2 x^4} \, dx\\ &=\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {9}{4};-\frac {1}{4},-\frac {1}{4};-\frac {5}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {1}{4};-\frac {1}{4},-\frac {1}{4};\frac {3}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} F_1\left (\frac {7}{4};-\frac {1}{4},-\frac {1}{4};\frac {11}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+2 \int \left (-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (i-\sqrt [4]{2} x^2\right )}+\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (i+\sqrt [4]{2} x^2\right )}\right ) \, dx+2 \int \left (-\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (1-\sqrt [4]{2} x^2\right )}+\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2\ 2^{3/4} \left (1+\sqrt [4]{2} x^2\right )}\right ) \, dx\\ &=\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {9}{4};-\frac {1}{4},-\frac {1}{4};-\frac {5}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {1}{4};-\frac {1}{4},-\frac {1}{4};\frac {3}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} F_1\left (\frac {7}{4};-\frac {1}{4},-\frac {1}{4};\frac {11}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i-\sqrt [4]{2} x^2} \, dx}{2^{3/4}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1-\sqrt [4]{2} x^2} \, dx}{2^{3/4}}+\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i+\sqrt [4]{2} x^2} \, dx}{2^{3/4}}+\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1+\sqrt [4]{2} x^2} \, dx}{2^{3/4}}\\ &=\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {9}{4};-\frac {1}{4},-\frac {1}{4};-\frac {5}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {1}{4};-\frac {1}{4},-\frac {1}{4};\frac {3}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} F_1\left (\frac {7}{4};-\frac {1}{4},-\frac {1}{4};\frac {11}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\int \left (\frac {i \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (i-\sqrt [8]{2} x\right )}+\frac {i \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (i+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}}-\frac {\int \left (\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (1-\sqrt [8]{2} x\right )}+\frac {\sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (1+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}}-\frac {\int \left (-\frac {(-1)^{3/4} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (\sqrt [4]{-1}-\sqrt [8]{2} x\right )}-\frac {(-1)^{3/4} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (\sqrt [4]{-1}+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}}+\frac {\int \left (-\frac {\sqrt [4]{-1} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (-(-1)^{3/4}-\sqrt [8]{2} x\right )}-\frac {\sqrt [4]{-1} \sqrt [4]{-1-2 x^4+2 x^8}}{2 \left (-(-1)^{3/4}+\sqrt [8]{2} x\right )}\right ) \, dx}{2^{3/4}}\\ &=\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {9}{4};-\frac {1}{4},-\frac {1}{4};-\frac {5}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{9 x^9 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {\sqrt [4]{-1-2 x^4+2 x^8} F_1\left (-\frac {1}{4};-\frac {1}{4},-\frac {1}{4};\frac {3}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {4 x^7 \sqrt [4]{-1-2 x^4+2 x^8} F_1\left (\frac {7}{4};-\frac {1}{4},-\frac {1}{4};\frac {11}{4};\frac {2 x^4}{1+\sqrt {3}},\frac {2 x^4}{1-\sqrt {3}}\right )}{7 \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {3}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {3}}}}+\frac {i \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i-\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}+\frac {i \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{i+\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1-\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{1+\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\sqrt [4]{-1} \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{-(-1)^{3/4}-\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}-\frac {\sqrt [4]{-1} \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{-(-1)^{3/4}+\sqrt [8]{2} x} \, dx}{2\ 2^{3/4}}+-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt [4]{-1}-\sqrt [8]{2} x} \, dx}{\sqrt [4]{2}}+-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \int \frac {\sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt [4]{-1}+\sqrt [8]{2} x} \, dx}{\sqrt [4]{2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.56, size = 160, normalized size = 0.92 \begin {gather*} \frac {1}{180} \left (\frac {4 \sqrt [4]{-1-2 x^4+2 x^8} \left (5+2 x^4+9 x^8-4 x^{12}+20 x^{16}\right )}{x^9}-45\ 2^{3/4} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt {2} x^2-\sqrt {-1-2 x^4+2 x^8}}\right )-45\ 2^{3/4} \tanh ^{-1}\left (\frac {2 x \sqrt [4]{-2-4 x^4+4 x^8}}{2 x^2+\sqrt {-2-4 x^4+4 x^8}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x^8)*(-1 - 2*x^4 + 2*x^8)^(1/4)*(1 - 3*x^8 + 4*x^16))/(x^10*(-1 + 2*x^8)),x]

[Out]

((4*(-1 - 2*x^4 + 2*x^8)^(1/4)*(5 + 2*x^4 + 9*x^8 - 4*x^12 + 20*x^16))/x^9 - 45*2^(3/4)*ArcTan[(2^(3/4)*x*(-1
- 2*x^4 + 2*x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[-1 - 2*x^4 + 2*x^8])] - 45*2^(3/4)*ArcTanh[(2*x*(-2 - 4*x^4 + 4*x^
8)^(1/4))/(2*x^2 + Sqrt[-2 - 4*x^4 + 4*x^8])])/180

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 131.49, size = 347, normalized size = 2.01

method	result	size
trager	\(\frac {\left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{16}-4 x^{12}+9 x^{8}+2 x^{4}+5\right )}{45 x^{9}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{8}+4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}-4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \sqrt {2 x^{8}-2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{2 x^{8}-1}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{8}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{4}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \sqrt {2 x^{8}-2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{2}+4 \left (2 x^{8}-2 x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{2 x^{8}-1}\right )}{4}\)	\(347\)
risch	\(\text {Expression too large to display}\)	\(1127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x,method=_RETURNVERBOSE)

[Out]

1/45*(2*x^8-2*x^4-1)^(1/4)*(20*x^16-4*x^12+9*x^8+2*x^4+5)/x^9+1/4*RootOf(_Z^2+RootOf(_Z^4+2)^2)*ln(-(-2*RootOf
(_Z^2+RootOf(_Z^4+2)^2)*RootOf(_Z^4+2)^2*x^8+4*RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^4-4*(2*x^8-2*x
^4-1)^(1/4)*RootOf(_Z^4+2)^2*x^3-4*(2*x^8-2*x^4-1)^(1/2)*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^2+4*(2*x^8-2*x^4-1)^(
3/4)*x+RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2))/(2*x^8-1))+1/4*RootOf(_Z^4+2)*ln(-(2*RootOf(_Z^4+2)^3*x
^8-4*RootOf(_Z^4+2)^3*x^4+4*(2*x^8-2*x^4-1)^(1/4)*RootOf(_Z^4+2)^2*x^3-4*(2*x^8-2*x^4-1)^(1/2)*RootOf(_Z^4+2)*
x^2+4*(2*x^8-2*x^4-1)^(3/4)*x-RootOf(_Z^4+2)^3)/(2*x^8-1))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x, algorithm="maxima")

[Out]

integrate((4*x^16 - 3*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(2*x^8 + 1)/((2*x^8 - 1)*x^10), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 982 vs. \(2 (145) = 290\).
time = 88.62, size = 982, normalized size = 5.68 \begin {gather*} -\frac {180 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \arctan \left (-\frac {32 \, x^{16} - 32 \, x^{8} + 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{9} - 8 \, x^{5} - x\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} + 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (6 \, x^{11} - 8 \, x^{7} - 3 \, x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} {\left (2 \, x^{10} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} - \sqrt {2} {\left (128 \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x^{5} + 8^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{16} - 40 \, x^{12} + 28 \, x^{8} + 20 \, x^{4} + 1\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{10} - 8 \, x^{6} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} + 32 \, {\left (2 \, x^{11} - x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x + 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} + \sqrt {2} {\left (2 \, x^{8} - 1\right )}}{2 \, x^{8} - 1}} + 8}{8 \, {\left (4 \, x^{16} - 64 \, x^{12} + 60 \, x^{8} + 32 \, x^{4} + 1\right )}}\right ) - 180 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \arctan \left (-\frac {32 \, x^{16} - 32 \, x^{8} - 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{9} - 8 \, x^{5} - x\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} - 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (6 \, x^{11} - 8 \, x^{7} - 3 \, x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} {\left (2 \, x^{10} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} - \sqrt {2} {\left (128 \, \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x^{5} - 8^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{16} - 40 \, x^{12} + 28 \, x^{8} + 20 \, x^{4} + 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{10} - 8 \, x^{6} - x^{2}\right )} \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} + 32 \, {\left (2 \, x^{11} - x^{3}\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {-\frac {8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} - \sqrt {2} {\left (2 \, x^{8} - 1\right )}}{2 \, x^{8} - 1}} + 8}{8 \, {\left (4 \, x^{16} - 64 \, x^{12} + 60 \, x^{8} + 32 \, x^{4} + 1\right )}}\right ) + 45 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x + 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} + \sqrt {2} {\left (2 \, x^{8} - 1\right )}\right )}}{2 \, x^{8} - 1}\right ) - 45 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{9} \log \left (-\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 8 \, \sqrt {2 \, x^{8} - 2 \, x^{4} - 1} x^{2} - \sqrt {2} {\left (2 \, x^{8} - 1\right )}\right )}}{2 \, x^{8} - 1}\right ) - 64 \, {\left (20 \, x^{16} - 4 \, x^{12} + 9 \, x^{8} + 2 \, x^{4} + 5\right )} {\left (2 \, x^{8} - 2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{2880 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x, algorithm="fricas")

[Out]

-1/2880*(180*8^(3/4)*sqrt(2)*x^9*arctan(-1/8*(32*x^16 - 32*x^8 + 4*8^(3/4)*sqrt(2)*(2*x^9 - 8*x^5 - x)*(2*x^8
- 2*x^4 - 1)^(3/4) + 16*8^(1/4)*sqrt(2)*(6*x^11 - 8*x^7 - 3*x^3)*(2*x^8 - 2*x^4 - 1)^(1/4) + 32*sqrt(2)*(2*x^1
0 - x^2)*sqrt(2*x^8 - 2*x^4 - 1) - sqrt(2)*(128*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(3/4)*x^5 + 8^(3/4)*sqrt(2)*(4*x^1
6 - 40*x^12 + 28*x^8 + 20*x^4 + 1) + 8*8^(1/4)*sqrt(2)*(2*x^10 - 8*x^6 - x^2)*sqrt(2*x^8 - 2*x^4 - 1) + 32*(2*
x^11 - x^3)*(2*x^8 - 2*x^4 - 1)^(1/4))*sqrt((8^(3/4)*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(1/4)*x^3 + 2*8^(1/4)*sqrt(2)
*(2*x^8 - 2*x^4 - 1)^(3/4)*x + 8*sqrt(2*x^8 - 2*x^4 - 1)*x^2 + sqrt(2)*(2*x^8 - 1))/(2*x^8 - 1)) + 8)/(4*x^16
- 64*x^12 + 60*x^8 + 32*x^4 + 1)) - 180*8^(3/4)*sqrt(2)*x^9*arctan(-1/8*(32*x^16 - 32*x^8 - 4*8^(3/4)*sqrt(2)*
(2*x^9 - 8*x^5 - x)*(2*x^8 - 2*x^4 - 1)^(3/4) - 16*8^(1/4)*sqrt(2)*(6*x^11 - 8*x^7 - 3*x^3)*(2*x^8 - 2*x^4 - 1
)^(1/4) + 32*sqrt(2)*(2*x^10 - x^2)*sqrt(2*x^8 - 2*x^4 - 1) - sqrt(2)*(128*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(3/4)*x
^5 - 8^(3/4)*sqrt(2)*(4*x^16 - 40*x^12 + 28*x^8 + 20*x^4 + 1) - 8*8^(1/4)*sqrt(2)*(2*x^10 - 8*x^6 - x^2)*sqrt(
2*x^8 - 2*x^4 - 1) + 32*(2*x^11 - x^3)*(2*x^8 - 2*x^4 - 1)^(1/4))*sqrt(-(8^(3/4)*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(
1/4)*x^3 + 2*8^(1/4)*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(3/4)*x - 8*sqrt(2*x^8 - 2*x^4 - 1)*x^2 - sqrt(2)*(2*x^8 - 1)
)/(2*x^8 - 1)) + 8)/(4*x^16 - 64*x^12 + 60*x^8 + 32*x^4 + 1)) + 45*8^(3/4)*sqrt(2)*x^9*log(8*(8^(3/4)*sqrt(2)*
(2*x^8 - 2*x^4 - 1)^(1/4)*x^3 + 2*8^(1/4)*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(3/4)*x + 8*sqrt(2*x^8 - 2*x^4 - 1)*x^2
+ sqrt(2)*(2*x^8 - 1))/(2*x^8 - 1)) - 45*8^(3/4)*sqrt(2)*x^9*log(-8*(8^(3/4)*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(1/4)
*x^3 + 2*8^(1/4)*sqrt(2)*(2*x^8 - 2*x^4 - 1)^(3/4)*x - 8*sqrt(2*x^8 - 2*x^4 - 1)*x^2 - sqrt(2)*(2*x^8 - 1))/(2
*x^8 - 1)) - 64*(20*x^16 - 4*x^12 + 9*x^8 + 2*x^4 + 5)*(2*x^8 - 2*x^4 - 1)^(1/4))/x^9

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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((2*x**8+1)*(2*x**8-2*x**4-1)**(1/4)*(4*x**16-3*x**8+1)/x**10/(2*x**8-1),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8+1)*(2*x^8-2*x^4-1)^(1/4)*(4*x^16-3*x^8+1)/x^10/(2*x^8-1),x, algorithm="giac")

[Out]

integrate((4*x^16 - 3*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(2*x^8 + 1)/((2*x^8 - 1)*x^10), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^8+1\right )\,{\left (2\,x^8-2\,x^4-1\right )}^{1/4}\,\left (4\,x^{16}-3\,x^8+1\right )}{x^{10}\,\left (2\,x^8-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(4*x^16 - 3*x^8 + 1))/(x^10*(2*x^8 - 1)),x)

[Out]

int(((2*x^8 + 1)*(2*x^8 - 2*x^4 - 1)^(1/4)*(4*x^16 - 3*x^8 + 1))/(x^10*(2*x^8 - 1)), x)

________________________________________________________________________________________