3.19.19 \(\int \frac {(-1-x-x^2+x^4) (2+x+2 x^4)}{\sqrt {-1-x+x^4} (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8)} \, dx\)

Optimal. Leaf size=123 \[ \frac {\sqrt [4]{-1} \sqrt {\sqrt {15}+7 i} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {15}}{2}} x}{2 \sqrt {x^4-x-1}}\right )}{\sqrt {10}}-(-1)^{3/4} \sqrt {\frac {1}{10} \left (\sqrt {15}-7 i\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {15}}{2}} x}{2 \sqrt {x^4-x-1}}\right ) \]

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Rubi [F]  time = 3.04, 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-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 - x - x^2 + x^4)*(2 + x + 2*x^4))/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 +
4*x^8)),x]

[Out]

Defer[Int][1/Sqrt[-1 - x + x^4], x]/2 - 4*Defer[Int][1/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*
x^5 + x^6 + 4*x^8)), x] - 7*Defer[Int][x/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*
x^8)), x] - (9*Defer[Int][x^2/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x])/
2 - Defer[Int][x^3/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x]/2 + (7*Defer
[Int][x^4/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x])/2 + 3*Defer[Int][x^5
/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x] - (5*Defer[Int][x^6/(Sqrt[-1 -
x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x])/2

Rubi steps

\begin {align*} \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx &=\int \left (\frac {1}{2 \sqrt {-1-x+x^4}}-\frac {8+14 x+9 x^2+x^3-7 x^4-6 x^5+5 x^6}{2 \sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \frac {8+14 x+9 x^2+x^3-7 x^4-6 x^5+5 x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx\\ &=\frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \left (\frac {8}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {14 x}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {9 x^2}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {x^3}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}-\frac {7 x^4}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}-\frac {6 x^5}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {5 x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \frac {x^3}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-\frac {5}{2} \int \frac {x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx+3 \int \frac {x^5}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx+\frac {7}{2} \int \frac {x^4}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-4 \int \frac {1}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-\frac {9}{2} \int \frac {x^2}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-7 \int \frac {x}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.89, size = 109075, normalized size = 886.79 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 - x - x^2 + x^4)*(2 + x + 2*x^4))/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 +
x^6 + 4*x^8)),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 3.76, size = 117, normalized size = 0.95 \begin {gather*} -(-1)^{3/4} \sqrt {\frac {1}{10} \left (-7 i+\sqrt {15}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{8}-\frac {i \sqrt {15}}{8}} x}{\sqrt {-1-x+x^4}}\right )+\frac {\sqrt [4]{-1} \sqrt {7 i+\sqrt {15}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{8}+\frac {i \sqrt {15}}{8}} x}{\sqrt {-1-x+x^4}}\right )}{\sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 - x - x^2 + x^4)*(2 + x + 2*x^4))/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 -
8*x^5 + x^6 + 4*x^8)),x]

[Out]

-((-1)^(3/4)*Sqrt[(-7*I + Sqrt[15])/10]*ArcTan[(Sqrt[1/8 - (I/8)*Sqrt[15]]*x)/Sqrt[-1 - x + x^4]]) + ((-1)^(1/
4)*Sqrt[7*I + Sqrt[15]]*ArcTan[(Sqrt[1/8 + (I/8)*Sqrt[15]]*x)/Sqrt[-1 - x + x^4]])/Sqrt[10]

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fricas [B]  time = 1.17, size = 935, normalized size = 7.60 \begin {gather*} \frac {1}{300} \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {3} \sqrt {2} \arctan \left (\frac {80 \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {3} \sqrt {2} {\left (2 \, x^{5} - x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt {x^{4} - x - 1} + 225 \, \sqrt {5} \sqrt {3} {\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} + 2 \, \sqrt {10} {\left (\sqrt {15} \sqrt {10} \sqrt {5} \sqrt {3} \sqrt {2} {\left (4 \, x^{8} - 9 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 9 \, x^{3} + 13 \, x^{2} + 8 \, x + 4\right )} + 120 \, \sqrt {5} \sqrt {3} {\left (x^{5} + x^{3} - x^{2} - x\right )} \sqrt {x^{4} - x - 1}\right )} \sqrt {\frac {20 \, x^{8} + 35 \, x^{6} - 40 \, x^{5} - 35 \, x^{4} + \sqrt {15} \sqrt {10} \sqrt {2} {\left (2 \, x^{5} + x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt {x^{4} - x - 1} - 35 \, x^{3} - 15 \, x^{2} + 40 \, x + 20}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}}}{375 \, {\left (4 \, x^{8} - 31 \, x^{6} - 8 \, x^{5} - 7 \, x^{4} + 31 \, x^{3} + 35 \, x^{2} + 8 \, x + 4\right )}}\right ) - \frac {1}{300} \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {3} \sqrt {2} \arctan \left (-\frac {80 \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {3} \sqrt {2} {\left (2 \, x^{5} - x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt {x^{4} - x - 1} - 225 \, \sqrt {5} \sqrt {3} {\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} + 2 \, \sqrt {10} {\left (\sqrt {15} \sqrt {10} \sqrt {5} \sqrt {3} \sqrt {2} {\left (4 \, x^{8} - 9 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 9 \, x^{3} + 13 \, x^{2} + 8 \, x + 4\right )} - 120 \, \sqrt {5} \sqrt {3} {\left (x^{5} + x^{3} - x^{2} - x\right )} \sqrt {x^{4} - x - 1}\right )} \sqrt {\frac {20 \, x^{8} + 35 \, x^{6} - 40 \, x^{5} - 35 \, x^{4} - \sqrt {15} \sqrt {10} \sqrt {2} {\left (2 \, x^{5} + x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt {x^{4} - x - 1} - 35 \, x^{3} - 15 \, x^{2} + 40 \, x + 20}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}}}{375 \, {\left (4 \, x^{8} - 31 \, x^{6} - 8 \, x^{5} - 7 \, x^{4} + 31 \, x^{3} + 35 \, x^{2} + 8 \, x + 4\right )}}\right ) - \frac {1}{80} \, \sqrt {15} \sqrt {10} \sqrt {2} \log \left (\frac {640 \, {\left (20 \, x^{8} + 35 \, x^{6} - 40 \, x^{5} - 35 \, x^{4} + \sqrt {15} \sqrt {10} \sqrt {2} {\left (2 \, x^{5} + x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt {x^{4} - x - 1} - 35 \, x^{3} - 15 \, x^{2} + 40 \, x + 20\right )}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) + \frac {1}{80} \, \sqrt {15} \sqrt {10} \sqrt {2} \log \left (\frac {640 \, {\left (20 \, x^{8} + 35 \, x^{6} - 40 \, x^{5} - 35 \, x^{4} - \sqrt {15} \sqrt {10} \sqrt {2} {\left (2 \, x^{5} + x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt {x^{4} - x - 1} - 35 \, x^{3} - 15 \, x^{2} + 40 \, x + 20\right )}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x, algorithm="fric
as")

[Out]

1/300*sqrt(15)*sqrt(10)*sqrt(5)*sqrt(3)*sqrt(2)*arctan(1/375*(80*sqrt(15)*sqrt(10)*sqrt(5)*sqrt(3)*sqrt(2)*(2*
x^5 - x^3 - 2*x^2 - 2*x)*sqrt(x^4 - x - 1) + 225*sqrt(5)*sqrt(3)*(4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 +
8*x + 4) + 2*sqrt(10)*(sqrt(15)*sqrt(10)*sqrt(5)*sqrt(3)*sqrt(2)*(4*x^8 - 9*x^6 - 8*x^5 - 9*x^4 + 9*x^3 + 13*x
^2 + 8*x + 4) + 120*sqrt(5)*sqrt(3)*(x^5 + x^3 - x^2 - x)*sqrt(x^4 - x - 1))*sqrt((20*x^8 + 35*x^6 - 40*x^5 -
35*x^4 + sqrt(15)*sqrt(10)*sqrt(2)*(2*x^5 + x^3 - 2*x^2 - 2*x)*sqrt(x^4 - x - 1) - 35*x^3 - 15*x^2 + 40*x + 20
)/(4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)))/(4*x^8 - 31*x^6 - 8*x^5 - 7*x^4 + 31*x^3 + 35*x^2 +
8*x + 4)) - 1/300*sqrt(15)*sqrt(10)*sqrt(5)*sqrt(3)*sqrt(2)*arctan(-1/375*(80*sqrt(15)*sqrt(10)*sqrt(5)*sqrt(3
)*sqrt(2)*(2*x^5 - x^3 - 2*x^2 - 2*x)*sqrt(x^4 - x - 1) - 225*sqrt(5)*sqrt(3)*(4*x^8 + x^6 - 8*x^5 - 7*x^4 - x
^3 + 3*x^2 + 8*x + 4) + 2*sqrt(10)*(sqrt(15)*sqrt(10)*sqrt(5)*sqrt(3)*sqrt(2)*(4*x^8 - 9*x^6 - 8*x^5 - 9*x^4 +
9*x^3 + 13*x^2 + 8*x + 4) - 120*sqrt(5)*sqrt(3)*(x^5 + x^3 - x^2 - x)*sqrt(x^4 - x - 1))*sqrt((20*x^8 + 35*x^
6 - 40*x^5 - 35*x^4 - sqrt(15)*sqrt(10)*sqrt(2)*(2*x^5 + x^3 - 2*x^2 - 2*x)*sqrt(x^4 - x - 1) - 35*x^3 - 15*x^
2 + 40*x + 20)/(4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)))/(4*x^8 - 31*x^6 - 8*x^5 - 7*x^4 + 31*x^
3 + 35*x^2 + 8*x + 4)) - 1/80*sqrt(15)*sqrt(10)*sqrt(2)*log(640*(20*x^8 + 35*x^6 - 40*x^5 - 35*x^4 + sqrt(15)*
sqrt(10)*sqrt(2)*(2*x^5 + x^3 - 2*x^2 - 2*x)*sqrt(x^4 - x - 1) - 35*x^3 - 15*x^2 + 40*x + 20)/(4*x^8 + x^6 - 8
*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)) + 1/80*sqrt(15)*sqrt(10)*sqrt(2)*log(640*(20*x^8 + 35*x^6 - 40*x^5 - 35
*x^4 - sqrt(15)*sqrt(10)*sqrt(2)*(2*x^5 + x^3 - 2*x^2 - 2*x)*sqrt(x^4 - x - 1) - 35*x^3 - 15*x^2 + 40*x + 20)/
(4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} + x + 2\right )} {\left (x^{4} - x^{2} - x - 1\right )}}{{\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} \sqrt {x^{4} - x - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x, algorithm="giac
")

[Out]

integrate((2*x^4 + x + 2)*(x^4 - x^2 - x - 1)/((4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)*sqrt(x^4
- x - 1)), x)

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maple [C]  time = 11.96, size = 709, normalized size = 5.76

method	result	size
trager	\(-5 \ln \left (\frac {-400 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{5} x^{2}+80 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{4}-120 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{2}+36 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{4}+160 \sqrt {x^{4}-x -1}\, \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x -80 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x -80 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}+27 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{2}-56 \sqrt {x^{4}-x -1}\, x -36 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x -36 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{20 x^{2} \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+4 x^{4}-3 x^{2}-4 x -4}\right ) \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}+\frac {7 \ln \left (\frac {-400 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{5} x^{2}+80 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{4}-120 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{2}+36 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{4}+160 \sqrt {x^{4}-x -1}\, \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x -80 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x -80 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}+27 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{2}-56 \sqrt {x^{4}-x -1}\, x -36 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x -36 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{20 x^{2} \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+4 x^{4}-3 x^{2}-4 x -4}\right ) \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{4}-\RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) \ln \left (-\frac {-50 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{4} x^{2}-10 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x^{4}+25 x^{2} \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+3 x^{4}+5 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) \sqrt {x^{4}-x -1}\, x +10 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x +10 \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}-3 x^{2}-3 x -3}{5 x^{2} \RootOf \left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}-x^{4}-x^{2}+x +1}\right )\)	\(709\)
default	\(\text {Expression too large to display}\)	\(8123\)
elliptic	\(\text {Expression too large to display}\)	\(8123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x,method=_RETURNVERBOSE)

[Out]

-5*ln((-400*RootOf(100*_Z^4-35*_Z^2+4)^5*x^2+80*RootOf(100*_Z^4-35*_Z^2+4)^3*x^4-120*RootOf(100*_Z^4-35*_Z^2+4
)^3*x^2+36*RootOf(100*_Z^4-35*_Z^2+4)*x^4+160*(x^4-x-1)^(1/2)*RootOf(100*_Z^4-35*_Z^2+4)^2*x-80*RootOf(100*_Z^
4-35*_Z^2+4)^3*x-80*RootOf(100*_Z^4-35*_Z^2+4)^3+27*RootOf(100*_Z^4-35*_Z^2+4)*x^2-56*(x^4-x-1)^(1/2)*x-36*Roo
tOf(100*_Z^4-35*_Z^2+4)*x-36*RootOf(100*_Z^4-35*_Z^2+4))/(20*x^2*RootOf(100*_Z^4-35*_Z^2+4)^2+4*x^4-3*x^2-4*x-
4))*RootOf(100*_Z^4-35*_Z^2+4)^3+7/4*ln((-400*RootOf(100*_Z^4-35*_Z^2+4)^5*x^2+80*RootOf(100*_Z^4-35*_Z^2+4)^3
*x^4-120*RootOf(100*_Z^4-35*_Z^2+4)^3*x^2+36*RootOf(100*_Z^4-35*_Z^2+4)*x^4+160*(x^4-x-1)^(1/2)*RootOf(100*_Z^
4-35*_Z^2+4)^2*x-80*RootOf(100*_Z^4-35*_Z^2+4)^3*x-80*RootOf(100*_Z^4-35*_Z^2+4)^3+27*RootOf(100*_Z^4-35*_Z^2+
4)*x^2-56*(x^4-x-1)^(1/2)*x-36*RootOf(100*_Z^4-35*_Z^2+4)*x-36*RootOf(100*_Z^4-35*_Z^2+4))/(20*x^2*RootOf(100*
_Z^4-35*_Z^2+4)^2+4*x^4-3*x^2-4*x-4))*RootOf(100*_Z^4-35*_Z^2+4)-RootOf(100*_Z^4-35*_Z^2+4)*ln(-(-50*RootOf(10
0*_Z^4-35*_Z^2+4)^4*x^2-10*RootOf(100*_Z^4-35*_Z^2+4)^2*x^4+25*x^2*RootOf(100*_Z^4-35*_Z^2+4)^2+3*x^4+5*RootOf
(100*_Z^4-35*_Z^2+4)*(x^4-x-1)^(1/2)*x+10*RootOf(100*_Z^4-35*_Z^2+4)^2*x+10*RootOf(100*_Z^4-35*_Z^2+4)^2-3*x^2
-3*x-3)/(5*x^2*RootOf(100*_Z^4-35*_Z^2+4)^2-x^4-x^2+x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} + x + 2\right )} {\left (x^{4} - x^{2} - x - 1\right )}}{{\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} \sqrt {x^{4} - x - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x, algorithm="maxi
ma")

[Out]

integrate((2*x^4 + x + 2)*(x^4 - x^2 - x - 1)/((4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)*sqrt(x^4
- x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (2\,x^4+x+2\right )\,\left (-x^4+x^2+x+1\right )}{\sqrt {x^4-x-1}\,\left (4\,x^8+x^6-8\,x^5-7\,x^4-x^3+3\,x^2+8\,x+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x + 2*x^4 + 2)*(x + x^2 - x^4 + 1))/((x^4 - x - 1)^(1/2)*(8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x
^8 + 4)),x)

[Out]

int(-((x + 2*x^4 + 2)*(x + x^2 - x^4 + 1))/((x^4 - x - 1)^(1/2)*(8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x
^8 + 4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x + 1\right ) \left (x^{3} - x^{2} - 1\right ) \left (2 x^{4} + x + 2\right )}{\sqrt {x^{4} - x - 1} \left (4 x^{8} + x^{6} - 8 x^{5} - 7 x^{4} - x^{3} + 3 x^{2} + 8 x + 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x**2-x-1)*(2*x**4+x+2)/(x**4-x-1)**(1/2)/(4*x**8+x**6-8*x**5-7*x**4-x**3+3*x**2+8*x+4),x)

[Out]

Integral((x + 1)*(x**3 - x**2 - 1)*(2*x**4 + x + 2)/(sqrt(x**4 - x - 1)*(4*x**8 + x**6 - 8*x**5 - 7*x**4 - x**
3 + 3*x**2 + 8*x + 4)), x)

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