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

Optimal. Leaf size=139 \[ -\frac {x}{2 \sqrt [4]{x^8-x^4-1}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^8-x^4-1}}{\sqrt {2} x^2-\sqrt {x^8-x^4-1}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{x^8-x^4-1}}{2 x^2+\sqrt {2} \sqrt {x^8-x^4-1}}\right )}{4\ 2^{3/4}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.33, 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^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(x*(1 - (2*x^4)/(1 - Sqrt[5]))^(1/4)*(1 - (2*x^4)/(1 + Sqrt[5]))^(1/4)*AppellF1[1/4, 1/4, 1/4, 5/4, (2*x^4)/(1
+ Sqrt[5]), (2*x^4)/(1 - Sqrt[5])])/(-1 - x^4 + x^8)^(1/4) - (x*(Sqrt[3 + Sqrt[5]] - Sqrt[2]*x^4)^(1/4)*(1 +
Sqrt[(3 + Sqrt[5])/2]*x^4)^(1/4)*AppellF1[1/4, 5/4, 1/4, 5/4, Sqrt[2/(3 + Sqrt[5])]*x^4, -(Sqrt[(3 + Sqrt[5])/
2]*x^4)])/(2*(3 + Sqrt[5])^(1/8)*(-1 - x^4 + x^8)^(1/4)) - (x*(Sqrt[3 - Sqrt[5]] + Sqrt[2]*x^4)^(1/4)*(1 - Sqr
t[(3 - Sqrt[5])/2]*x^4)^(1/4)*AppellF1[1/4, 5/4, 1/4, 5/4, -(Sqrt[(3 + Sqrt[5])/2]*x^4), Sqrt[(3 - Sqrt[5])/2]
*x^4])/(2*(3 - Sqrt[5])^(1/8)*(-1 - x^4 + x^8)^(1/4)) - (Sqrt[3 - Sqrt[5]]*Defer[Int][1/((Sqrt[3 - Sqrt[5]] -
Sqrt[2]*x^4)*(-1 - x^4 + x^8)^(1/4)), x])/2 - (Sqrt[3 + Sqrt[5]]*Defer[Int][1/((Sqrt[3 + Sqrt[5]] + Sqrt[2]*x^
4)*(-1 - x^4 + x^8)^(1/4)), x])/2

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 2.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.39, size = 139, normalized size = 1.00 \begin {gather*} -\frac {x}{2 \sqrt [4]{-1-x^4+x^8}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{-1-x^4+x^8}}{\sqrt {2} x^2-\sqrt {-1-x^4+x^8}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {-1-x^4+x^8}}\right )}{4\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*x/(-1 - x^4 + x^8)^(1/4) + ArcTan[(2^(3/4)*x*(-1 - x^4 + x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[-1 - x^4 + x^8])
]/(4*2^(3/4)) - ArcTanh[(2*2^(1/4)*x*(-1 - x^4 + x^8)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[-1 - x^4 + x^8])]/(4*2^(3/4
))

________________________________________________________________________________________

fricas [B]  time = 26.80, size = 658, normalized size = 4.73 \begin {gather*} -\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{8} + x^{4} - 1\right )}\right )} \sqrt {\frac {x^{8} + x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{8} + x^{4} - 1}}}{2 \, {\left (x^{8} - 3 \, x^{4} - 1\right )}}\right ) + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{8} + x^{4} - 1\right )}\right )} \sqrt {\frac {x^{8} + x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{8} + x^{4} - 1}}}{2 \, {\left (x^{8} - 3 \, x^{4} - 1\right )}}\right ) + 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \log \left (\frac {2 \, {\left (x^{8} + x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{8} + x^{4} - 1}\right ) - 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \log \left (\frac {2 \, {\left (x^{8} + x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{8} + x^{4} - 1}\right ) + 16 \, {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{8} - x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(4*2^(1/4)*(x^8 - x^4 - 1)*arctan(1/2*(4*2^(1/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 2*2^(3/4)*(x^8 - x^4 - 1)^(
3/4)*x + sqrt(2)*(2*2^(3/4)*(x^8 - x^4 - 1)^(1/4)*x^3 - 4*sqrt(x^8 - x^4 - 1)*x^2 + 2*2^(1/4)*(x^8 - x^4 - 1)^
(3/4)*x - sqrt(2)*(x^8 + x^4 - 1))*sqrt((x^8 + x^4 + 4*2^(1/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 4*sqrt(2)*sqrt(x^8
- x^4 - 1)*x^2 + 2*2^(3/4)*(x^8 - x^4 - 1)^(3/4)*x - 1)/(x^8 + x^4 - 1)))/(x^8 - 3*x^4 - 1)) + 4*2^(1/4)*(x^8
- x^4 - 1)*arctan(1/2*(4*2^(1/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 2*2^(3/4)*(x^8 - x^4 - 1)^(3/4)*x + sqrt(2)*(2*2^
(3/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 4*sqrt(x^8 - x^4 - 1)*x^2 + 2*2^(1/4)*(x^8 - x^4 - 1)^(3/4)*x + sqrt(2)*(x^8
+ x^4 - 1))*sqrt((x^8 + x^4 - 4*2^(1/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 4*sqrt(2)*sqrt(x^8 - x^4 - 1)*x^2 - 2*2^(
3/4)*(x^8 - x^4 - 1)^(3/4)*x - 1)/(x^8 + x^4 - 1)))/(x^8 - 3*x^4 - 1)) + 2^(1/4)*(x^8 - x^4 - 1)*log(2*(x^8 +
x^4 + 4*2^(1/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 4*sqrt(2)*sqrt(x^8 - x^4 - 1)*x^2 + 2*2^(3/4)*(x^8 - x^4 - 1)^(3/4
)*x - 1)/(x^8 + x^4 - 1)) - 2^(1/4)*(x^8 - x^4 - 1)*log(2*(x^8 + x^4 - 4*2^(1/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 4
*sqrt(2)*sqrt(x^8 - x^4 - 1)*x^2 - 2*2^(3/4)*(x^8 - x^4 - 1)^(3/4)*x - 1)/(x^8 + x^4 - 1)) + 16*(x^8 - x^4 - 1
)^(3/4)*x)/(x^8 - x^4 - 1)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

________________________________________________________________________________________

maple [C]  time = 7.93, size = 287, normalized size = 2.06

method	result	size
risch	\(-\frac {x}{2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {x^{8} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{8}-x^{4}-1}\, x^{2}+2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}+4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{x^{8}+x^{4}-1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{8}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{8}-x^{4}-1}\, x^{2}-2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{4}+4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+8\right )}{x^{8}+x^{4}-1}\right )}{16}\)	\(287\)
trager	\(-\frac {x}{2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {x^{8} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{8}-x^{4}-1}\, x^{2}-2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}-4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{x^{8}+x^{4}-1}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{8}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{8}-x^{4}-1}\, x^{2}-2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{4}+4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+8\right )}{x^{8}+x^{4}-1}\right )}{16}\)	\(288\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x/(x^8-x^4-1)^(1/4)-1/16*RootOf(_Z^2+RootOf(_Z^4+8)^2)*ln(-(x^8*RootOf(_Z^2+RootOf(_Z^4+8)^2)+RootOf(_Z^4
+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*(x^8-x^4-1)^(1/2)*x^2+2*(x^8-x^4-1)^(1/4)*RootOf(_Z^4+8)^2*x^3-3*RootOf(_Z
^2+RootOf(_Z^4+8)^2)*x^4+4*(x^8-x^4-1)^(3/4)*x-RootOf(_Z^2+RootOf(_Z^4+8)^2))/(x^8+x^4-1))+1/16*RootOf(_Z^4+8)
*ln(-(-RootOf(_Z^4+8)*x^8+RootOf(_Z^4+8)^3*(x^8-x^4-1)^(1/2)*x^2-2*(x^8-x^4-1)^(1/4)*RootOf(_Z^4+8)^2*x^3+3*Ro
otOf(_Z^4+8)*x^4+4*(x^8-x^4-1)^(3/4)*x+RootOf(_Z^4+8))/(x^8+x^4-1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + 1\right )} {\left (x^{8} - 1\right )}}{{\left (x^{16} - 3 \, x^{8} + 1\right )} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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**8-1)*(x**8+1)/(x**8-x**4-1)**(1/4)/(x**16-3*x**8+1),x)

[Out]

Timed out

________________________________________________________________________________________