Optimal. Leaf size=131 \[ -\tan ^{-1}\left (\frac {\sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\frac {1}{2} \sqrt {\frac {1}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\frac {1}{2} \sqrt {\frac {1}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right ) \]
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Rubi [C] time = 1.91, antiderivative size = 611, normalized size of antiderivative = 4.66, number of steps used = 27, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2056, 6725, 716, 1098, 934, 168, 538, 537} \begin {gather*} \frac {\sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \left (-1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 168
Rule 537
Rule 538
Rule 716
Rule 934
Rule 1098
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^4}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1+x^4}{\sqrt {x} \sqrt {-1-x+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {2}{\sqrt {x} \sqrt {-1-x+x^2} \left (-1+x^4\right )}\right ) \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1-x+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {1}{2 \sqrt {x} \left (1-x^2\right ) \sqrt {-1-x+x^2}}-\frac {1}{2 \sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1-x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {i}{2 (i-x) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {i}{2 \sqrt {x} (i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {1}{2 (1-x) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {1}{2 \sqrt {x} (1+x) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \left (-1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 8.75, size = 1848, normalized size = 14.11
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.30, size = 131, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {\sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 2485, normalized size = 18.97
result too large to display
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 {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x^{2} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.09, size = 913, normalized size = 6.97
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \ln \left (\frac {5498560 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4} x^{2}-8247840 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4} x -5498560 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4}+566112 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}-2017612 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x -566112 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right )+162560 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-x}+11529 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}-49959 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x -11529 \RootOf \left (\textit {\_Z}^{2}+100 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+10\right )-572030 \sqrt {x^{3}-x^{2}-x}}{\left (4 x \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-8 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x \right )^{2}}\right )}{20}+\frac {\RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {27492800 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5} x^{2}-41239200 x \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5}-27492800 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5}+2668000 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x^{2}+1840220 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x -2668000 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3}+81280 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-x}+49517 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}+346619 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x -49517 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )+294143 \sqrt {x^{3}-x^{2}-x}}{\left (20 x \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-40 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-3 x -4\right )^{2}}\right )}{2}+5 \ln \left (-\frac {20 x^{2} \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-40 x \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-20 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}-4 \sqrt {x^{3}-x^{2}-x}-2 x -1}{\left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+\frac {\ln \left (-\frac {20 x^{2} \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-40 x \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-20 \RootOf \left (80 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}-4 \sqrt {x^{3}-x^{2}-x}-2 x -1}{\left (-1+x \right ) \left (1+x \right )}\right )}{4}\) | \(913\) |
default | \(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}-\frac {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}+\frac {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\) | \(1009\) |
elliptic | \(\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{25 \sqrt {x^{3}-x^{2}-x}}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{50 \sqrt {x^{3}-x^{2}-x}}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{25 \sqrt {x^{3}-x^{2}-x}}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{50 \sqrt {x^{3}-x^{2}-x}}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\) | \(1685\) |
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 {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x^{2} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 658, normalized size = 5.02 \begin {gather*} \frac {2\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}-\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}}{2}-\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}-\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}}{2}+\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}-\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}-\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,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 {x^{4} + 1}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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