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.
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\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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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