Optimal. Leaf size=234 \[ \frac {1}{2} \tan ^{-1}\left (\frac {1}{2} \left (-1+\left (1+\frac {1}{x}\right )^2\right )\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )-\frac {1}{4} \sqrt {\frac {1}{5} \left (-2+\sqrt {5}\right )} \log \left (\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )+\frac {1}{4} \sqrt {\frac {1}{5} \left (-2+\sqrt {5}\right )} \log \left (\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right ) \]
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Rubi [A]
time = 0.24, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {2094, 1687,
1183, 648, 632, 210, 642, 12, 1121} \begin {gather*} \frac {1}{2} \text {ArcTan}\left (\frac {1}{2} \left (\left (\frac {1}{x}+1\right )^2-1\right )\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\frac {2}{x}-\sqrt {2 \left (1+\sqrt {5}\right )}+2}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\frac {2}{x}+\sqrt {2 \left (1+\sqrt {5}\right )}+2}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )-\frac {1}{4} \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \log \left (\left (\frac {1}{x}+1\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right )+\frac {1}{4} \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \log \left (\left (\frac {1}{x}+1\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1121
Rule 1183
Rule 1687
Rule 2094
Rubi steps
\begin {align*} \int \frac {1}{1+4 x+4 x^2+4 x^4} \, dx &=-\left (16 \text {Subst}\left (\int \frac {(4-4 x)^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )\right )\\ &=-\left (16 \text {Subst}\left (\int -\frac {32 x}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )\right )-16 \text {Subst}\left (\int \frac {16+16 x^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )\\ &=512 \text {Subst}\left (\int \frac {x}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )-\frac {\text {Subst}\left (\int \frac {16 \sqrt {2 \left (1+\sqrt {5}\right )}-\left (16-16 \sqrt {5}\right ) x}{\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{32 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\text {Subst}\left (\int \frac {16 \sqrt {2 \left (1+\sqrt {5}\right )}+\left (16-16 \sqrt {5}\right ) x}{\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{32 \sqrt {10 \left (1+\sqrt {5}\right )}}\\ &=256 \text {Subst}\left (\int \frac {1}{1280-512 x+256 x^2} \, dx,x,\left (1+\frac {1}{x}\right )^2\right )+\frac {\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {5}\right )}+2 x}{\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{4 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {5}\right )}+2 x}{\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{4 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {1}{20} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )-\frac {1}{20} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )\\ &=-\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )+\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )-512 \text {Subst}\left (\int \frac {1}{-1048576-x^2} \, dx,x,-512+512 \left (1+\frac {1}{x}\right )^2\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {5}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {5}\right )}+2 \left (1+\frac {1}{x}\right )\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {5}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {5}\right )}+2 \left (1+\frac {1}{x}\right )\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {1}{2} \left (-1+\left (1+\frac {1}{x}\right )^2\right )\right )-\frac {\left (1+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )}{4 \sqrt {10}}-\frac {\left (1+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )}{4 \sqrt {10}}-\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )+\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 47, normalized size = 0.20 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1+4 \text {$\#$1}+4 \text {$\#$1}^2+4 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{1+2 \text {$\#$1}+4 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.02, size = 41, normalized size = 0.18
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+2 \textit {\_R} +1}\right )}{4}\) | \(41\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+2 \textit {\_R} +1}\right )}{4}\) | \(41\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.13, size = 499, normalized size = 2.13 \begin {gather*} -\frac {1}{20} \, {\left (\sqrt {10} \sqrt {-\frac {15}{8} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )}^{2} - \frac {5}{4} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )} - \frac {15}{8} \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - 9} - 5 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - 5 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}}\right )} \log \left (\frac {5}{2} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )}^{2} {\left (12 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + 6 i - 1\right )} + 15 \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - \frac {5}{2} \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} + {\left ({\left (6 \, \sqrt {10} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )} - \sqrt {10}\right )} {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} - \sqrt {10} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}\right )} \sqrt {-\frac {15}{8} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )}^{2} - \frac {5}{4} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )} - \frac {15}{8} \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - 9} + 8 \, x + 3\right ) + \frac {1}{20} \, {\left (\sqrt {10} \sqrt {-\frac {15}{8} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )}^{2} - \frac {5}{4} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )} - \frac {15}{8} \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - 9} + 5 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} + 5 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}}\right )} \log \left (\frac {5}{2} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )}^{2} {\left (12 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + 6 i - 1\right )} + 15 \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - \frac {5}{2} \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - {\left ({\left (6 \, \sqrt {10} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )} - \sqrt {10}\right )} {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} - \sqrt {10} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}\right )} \sqrt {-\frac {15}{8} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )}^{2} - \frac {5}{4} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )} - \frac {15}{8} \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - 9} + 8 \, x + 3\right ) - \frac {1}{4} \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} \log \left (-5 \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )}^{2} {\left (12 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + 6 i - 1\right )} - 30 \, {\left (2 \, \sqrt {\frac {1}{10} i - \frac {1}{5}} - i\right )} {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} - 30 \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{3} + 8 \, x - 216 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} - 108 i + 21\right ) - \frac {1}{4} \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )} \log \left (30 \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{3} + 5 \, {\left (2 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + i\right )}^{2} + 8 \, x + 216 \, \sqrt {-\frac {1}{10} i - \frac {1}{5}} + 108 i - 27\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3432 vs.
\(2 (190) = 380\).
time = 1.36, size = 3432, normalized size = 14.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 4.06, size = 265, normalized size = 1.13 \begin {gather*} -\frac {1}{20} \, {\left (\left (i + 2\right ) \, \sqrt {\sqrt {5} - 2} {\left (\frac {i}{\sqrt {5} - 2} + 1\right )} + 5 i\right )} \log \left (\left (406 i + 174\right ) \, \sqrt {5} x + \left (868 i + 372\right ) \, x + 29 \, \sqrt {5} \sqrt {29 \, \sqrt {5} + 62} + \left (87 i - 203\right ) \, \sqrt {5} + \left (19 i + 62\right ) \, \sqrt {29 \, \sqrt {5} + 62} + 186 i - 434\right ) - \frac {1}{20} \, {\left (\left (i + 2\right ) \, \sqrt {\sqrt {5} - 2} {\left (-\frac {i}{\sqrt {5} - 2} - 1\right )} + 5 i\right )} \log \left (\left (406 i + 174\right ) \, \sqrt {5} x + \left (868 i + 372\right ) \, x - 29 \, \sqrt {5} \sqrt {29 \, \sqrt {5} + 62} + \left (87 i - 203\right ) \, \sqrt {5} - \left (19 i + 62\right ) \, \sqrt {29 \, \sqrt {5} + 62} + 186 i - 434\right ) - \frac {1}{20} \, {\left (\left (2 i + 1\right ) \, \sqrt {\sqrt {5} + 2} {\left (-\frac {i}{\sqrt {5} + 2} - 1\right )} - 5 i\right )} \log \left (\left (26 i + 130\right ) \, \sqrt {5} x - \left (44 i + 220\right ) \, x + 13 \, \sqrt {5} \sqrt {13 \, \sqrt {5} - 22} - \left (65 i - 13\right ) \, \sqrt {5} + \left (19 i - 22\right ) \, \sqrt {13 \, \sqrt {5} - 22} + 110 i - 22\right ) - \frac {1}{20} \, {\left (\left (2 i + 1\right ) \, \sqrt {\sqrt {5} + 2} {\left (\frac {i}{\sqrt {5} + 2} + 1\right )} - 5 i\right )} \log \left (\left (26 i + 130\right ) \, \sqrt {5} x - \left (44 i + 220\right ) \, x - 13 \, \sqrt {5} \sqrt {13 \, \sqrt {5} - 22} - \left (65 i - 13\right ) \, \sqrt {5} - \left (19 i - 22\right ) \, \sqrt {13 \, \sqrt {5} - 22} + 110 i - 22\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.36, size = 87, normalized size = 0.37 \begin {gather*} \sum _{k=1}^4\ln \left (-\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right )\,\left (\frac {x}{4}+\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right )\,\left (6\,x+\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right )\,\left (36\,x+16\right )\right )\right )\right )\,\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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