3.21.2 \(\int \frac {(2+x^6) (-1-x^4+x^6)}{\sqrt [4]{1-x^4-x^6} (-1+x^6)^2} \, dx\)

Optimal. Leaf size=141 \[ -\frac {\left (-x^6-x^4+1\right )^{3/4} x}{2 \left (x^6-1\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^6-x^4+1}}{\sqrt {-x^6-x^4+1}-x^2}\right )}{4 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^6-x^4+1}}{x^2+\sqrt {-x^6-x^4+1}}\right )}{4 \sqrt {2}} \]

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Rubi [F]  time = 2.74, 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 (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(1 - x^4 - x^6)^(-1/4), x] - Defer[Int][1/((-1 + I*Sqrt[3] - 2*x)^2*(1 - x^4 - x^6)^(1/4)), x]/3 +
((I/3)*Defer[Int][1/((-1 + I*Sqrt[3] - 2*x)*(1 - x^4 - x^6)^(1/4)), x])/Sqrt[3] - Defer[Int][1/((1 + I*Sqrt[3]
- 2*x)^2*(1 - x^4 - x^6)^(1/4)), x]/3 + ((I/3)*Defer[Int][1/((1 + I*Sqrt[3] - 2*x)*(1 - x^4 - x^6)^(1/4)), x]
)/Sqrt[3] - Defer[Int][1/((-1 + x)^2*(1 - x^4 - x^6)^(1/4)), x]/12 + (5*Defer[Int][1/((-1 + x)*(1 - x^4 - x^6)
^(1/4)), x])/12 - Defer[Int][1/((1 + x)^2*(1 - x^4 - x^6)^(1/4)), x]/12 - (5*Defer[Int][1/((1 + x)*(1 - x^4 -
x^6)^(1/4)), x])/12 + (5*(3 + (5*I)*Sqrt[3])*Defer[Int][1/((-1 - I*Sqrt[3] + 2*x)*(1 - x^4 - x^6)^(1/4)), x])/
36 - (5*(3 - (5*I)*Sqrt[3])*Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*(1 - x^4 - x^6)^(1/4)), x])/36 - Defer[Int][1/
((-1 + I*Sqrt[3] + 2*x)^2*(1 - x^4 - x^6)^(1/4)), x]/3 + ((I/3)*Defer[Int][1/((-1 + I*Sqrt[3] + 2*x)*(1 - x^4
- x^6)^(1/4)), x])/Sqrt[3] + (5*(3 - (5*I)*Sqrt[3])*Defer[Int][1/((-1 + I*Sqrt[3] + 2*x)*(1 - x^4 - x^6)^(1/4)
), x])/36 - Defer[Int][1/((1 + I*Sqrt[3] + 2*x)^2*(1 - x^4 - x^6)^(1/4)), x]/3 + ((I/3)*Defer[Int][1/((1 + I*S
qrt[3] + 2*x)*(1 - x^4 - x^6)^(1/4)), x])/Sqrt[3] - (5*(3 + (5*I)*Sqrt[3])*Defer[Int][1/((1 + I*Sqrt[3] + 2*x)
*(1 - x^4 - x^6)^(1/4)), x])/36

Rubi steps

\begin {align*} \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx &=\int \left (\frac {1}{\sqrt [4]{1-x^4-x^6}}-\frac {1}{12 (-1+x)^2 \sqrt [4]{1-x^4-x^6}}-\frac {1}{12 (1+x)^2 \sqrt [4]{1-x^4-x^6}}+\frac {5}{6 \left (-1+x^2\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1}{4 \left (1-x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {5 (-3+x)}{12 \left (1-x+x^2\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1}{4 \left (1+x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}}-\frac {5 (3+x)}{12 \left (1+x+x^2\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {1}{(-1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx\right )-\frac {1}{12} \int \frac {1}{(1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{4} \int \frac {1}{\left (1-x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{4} \int \frac {1}{\left (1+x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {5}{12} \int \frac {-3+x}{\left (1-x+x^2\right ) \sqrt [4]{1-x^4-x^6}} \, dx-\frac {5}{12} \int \frac {3+x}{\left (1+x+x^2\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {5}{6} \int \frac {1}{\left (-1+x^2\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\int \frac {1}{\sqrt [4]{1-x^4-x^6}} \, dx\\ &=-\left (\frac {1}{12} \int \frac {1}{(-1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx\right )-\frac {1}{12} \int \frac {1}{(1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{4} \int \left (-\frac {4}{3 \left (1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}}-\frac {4}{3 \left (-1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\frac {1}{4} \int \left (-\frac {4}{3 \left (-1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}}-\frac {4}{3 \left (1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\frac {5}{12} \int \left (\frac {1+\frac {5 i}{\sqrt {3}}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1-\frac {5 i}{\sqrt {3}}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx-\frac {5}{12} \int \left (\frac {1-\frac {5 i}{\sqrt {3}}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1+\frac {5 i}{\sqrt {3}}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\frac {5}{6} \int \left (\frac {1}{2 (-1+x) \sqrt [4]{1-x^4-x^6}}-\frac {1}{2 (1+x) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\int \frac {1}{\sqrt [4]{1-x^4-x^6}} \, dx\\ &=-\left (\frac {1}{12} \int \frac {1}{(-1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx\right )-\frac {1}{12} \int \frac {1}{(1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {5}{12} \int \frac {1}{(-1+x) \sqrt [4]{1-x^4-x^6}} \, dx-\frac {5}{12} \int \frac {1}{(1+x) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {i \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}+\frac {i \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}+\frac {i \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}+\frac {i \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}-\frac {1}{36} \left (5 \left (3-5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{36} \left (5 \left (3-5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{36} \left (5 \left (3+5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{36} \left (5 \left (3+5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\int \frac {1}{\sqrt [4]{1-x^4-x^6}} \, dx\\ \end {align*}

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Mathematica [F]  time = 1.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.85, size = 141, normalized size = 1.00 \begin {gather*} -\frac {x \left (1-x^4-x^6\right )^{3/4}}{2 \left (-1+x^6\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{-x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(x*(1 - x^4 - x^6)^(3/4))/(-1 + x^6) - (5*ArcTan[(Sqrt[2]*x*(1 - x^4 - x^6)^(1/4))/(-x^2 + Sqrt[1 - x^4 -
x^6])])/(4*Sqrt[2]) - (5*ArcTanh[(Sqrt[2]*x*(1 - x^4 - x^6)^(1/4))/(x^2 + Sqrt[1 - x^4 - x^6])])/(4*Sqrt[2])

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fricas [B]  time = 10.90, size = 852, normalized size = 6.04 \begin {gather*} -\frac {20 \, \sqrt {2} {\left (x^{6} - 1\right )} \arctan \left (-\frac {x^{12} - 2 \, x^{6} + 2 \, \sqrt {2} {\left (x^{7} + 4 \, x^{5} - x\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} + 4 \, x^{7} - 3 \, x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} - 4 \, {\left (x^{8} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} - {\left (16 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} - \sqrt {2} {\left (x^{12} + 10 \, x^{10} + 8 \, x^{8} - 2 \, x^{6} - 10 \, x^{4} + 1\right )} - 4 \, {\left (x^{9} - x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1}{x^{6} - 1}} + 1}{x^{12} + 16 \, x^{10} + 16 \, x^{8} - 2 \, x^{6} - 16 \, x^{4} + 1}\right ) - 20 \, \sqrt {2} {\left (x^{6} - 1\right )} \arctan \left (-\frac {x^{12} - 2 \, x^{6} - 2 \, \sqrt {2} {\left (x^{7} + 4 \, x^{5} - x\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} + 4 \, x^{7} - 3 \, x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} - 4 \, {\left (x^{8} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} - {\left (16 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} + \sqrt {2} {\left (x^{12} + 10 \, x^{10} + 8 \, x^{8} - 2 \, x^{6} - 10 \, x^{4} + 1\right )} - 4 \, {\left (x^{9} - x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1}{x^{6} - 1}} + 1}{x^{12} + 16 \, x^{10} + 16 \, x^{8} - 2 \, x^{6} - 16 \, x^{4} + 1}\right ) - 5 \, \sqrt {2} {\left (x^{6} - 1\right )} \log \left (\frac {4 \, {\left (x^{6} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{6} - 1}\right ) + 5 \, \sqrt {2} {\left (x^{6} - 1\right )} \log \left (\frac {4 \, {\left (x^{6} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{6} - 1}\right ) + 16 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{6} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(20*sqrt(2)*(x^6 - 1)*arctan(-(x^12 - 2*x^6 + 2*sqrt(2)*(x^7 + 4*x^5 - x)*(-x^6 - x^4 + 1)^(3/4) + 2*sqr
t(2)*(3*x^9 + 4*x^7 - 3*x^3)*(-x^6 - x^4 + 1)^(1/4) - 4*(x^8 - x^2)*sqrt(-x^6 - x^4 + 1) - (16*(-x^6 - x^4 + 1
)^(3/4)*x^5 + 2*sqrt(2)*(x^8 + 4*x^6 - x^2)*sqrt(-x^6 - x^4 + 1) - sqrt(2)*(x^12 + 10*x^10 + 8*x^8 - 2*x^6 - 1
0*x^4 + 1) - 4*(x^9 - x^3)*(-x^6 - x^4 + 1)^(1/4))*sqrt((x^6 + 2*sqrt(2)*(-x^6 - x^4 + 1)^(1/4)*x^3 - 4*sqrt(-
x^6 - x^4 + 1)*x^2 + 2*sqrt(2)*(-x^6 - x^4 + 1)^(3/4)*x - 1)/(x^6 - 1)) + 1)/(x^12 + 16*x^10 + 16*x^8 - 2*x^6
- 16*x^4 + 1)) - 20*sqrt(2)*(x^6 - 1)*arctan(-(x^12 - 2*x^6 - 2*sqrt(2)*(x^7 + 4*x^5 - x)*(-x^6 - x^4 + 1)^(3/
4) - 2*sqrt(2)*(3*x^9 + 4*x^7 - 3*x^3)*(-x^6 - x^4 + 1)^(1/4) - 4*(x^8 - x^2)*sqrt(-x^6 - x^4 + 1) - (16*(-x^6
- x^4 + 1)^(3/4)*x^5 - 2*sqrt(2)*(x^8 + 4*x^6 - x^2)*sqrt(-x^6 - x^4 + 1) + sqrt(2)*(x^12 + 10*x^10 + 8*x^8 -
2*x^6 - 10*x^4 + 1) - 4*(x^9 - x^3)*(-x^6 - x^4 + 1)^(1/4))*sqrt((x^6 - 2*sqrt(2)*(-x^6 - x^4 + 1)^(1/4)*x^3
- 4*sqrt(-x^6 - x^4 + 1)*x^2 - 2*sqrt(2)*(-x^6 - x^4 + 1)^(3/4)*x - 1)/(x^6 - 1)) + 1)/(x^12 + 16*x^10 + 16*x^
8 - 2*x^6 - 16*x^4 + 1)) - 5*sqrt(2)*(x^6 - 1)*log(4*(x^6 + 2*sqrt(2)*(-x^6 - x^4 + 1)^(1/4)*x^3 - 4*sqrt(-x^6
- x^4 + 1)*x^2 + 2*sqrt(2)*(-x^6 - x^4 + 1)^(3/4)*x - 1)/(x^6 - 1)) + 5*sqrt(2)*(x^6 - 1)*log(4*(x^6 - 2*sqrt
(2)*(-x^6 - x^4 + 1)^(1/4)*x^3 - 4*sqrt(-x^6 - x^4 + 1)*x^2 - 2*sqrt(2)*(-x^6 - x^4 + 1)^(3/4)*x - 1)/(x^6 - 1
)) + 16*(-x^6 - x^4 + 1)^(3/4)*x)/(x^6 - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 20.67, size = 308, normalized size = 2.18

method	result	size
trager	\(-\frac {x \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}}}{2 \left (x^{6}-1\right )}-\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {-x^{6}-x^{4}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{8}-\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {-x^{6}-x^{4}+1}\, x^{2}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{8}\)	\(308\)
risch	\(\frac {x \left (x^{6}+x^{4}-1\right )}{2 \left (x^{6}-1\right ) \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {-x^{6}-x^{4}+1}\, x^{2}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{8}+\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right ) x -2 x^{2} \sqrt {-x^{6}-x^{4}+1}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{8}\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x/(x^6-1)*(-x^6-x^4+1)^(3/4)-5/8*RootOf(_Z^4+1)*ln((-2*RootOf(_Z^4+1)^3*(-x^6-x^4+1)^(1/2)*x^2-RootOf(_Z^
4+1)*x^6-2*(-x^6-x^4+1)^(1/4)*RootOf(_Z^4+1)^2*x^3-2*RootOf(_Z^4+1)*x^4+2*(-x^6-x^4+1)^(3/4)*x+RootOf(_Z^4+1))
/(-1+x)/(1+x)/(x^2+x+1)/(x^2-x+1))-5/8*RootOf(_Z^4+1)^3*ln((-RootOf(_Z^4+1)^3*x^6-2*RootOf(_Z^4+1)^3*x^4+2*(-x
^6-x^4+1)^(1/4)*RootOf(_Z^4+1)^2*x^3-2*RootOf(_Z^4+1)*(-x^6-x^4+1)^(1/2)*x^2+2*(-x^6-x^4+1)^(3/4)*x+RootOf(_Z^
4+1)^3)/(-1+x)/(1+x)/(x^2+x+1)/(x^2-x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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