Optimal. Leaf size=317 \[ -\frac {17-\left (1+\frac {1}{x}\right )^2}{2 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {\left (59-17 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right )}{10 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {7}{4} \tan ^{-1}\left (\frac {1}{2} \left (-1+\left (1+\frac {1}{x}\right )^2\right )\right )-\frac {1}{20} \sqrt {\frac {1}{10} \left (5959+2665 \sqrt {5}\right )} \tan ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )-\frac {1}{20} \sqrt {\frac {1}{10} \left (5959+2665 \sqrt {5}\right )} \tan ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )+\frac {1}{40} \sqrt {\frac {1}{10} \left (-5959+2665 \sqrt {5}\right )} \log \left (\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )-\frac {1}{40} \sqrt {\frac {1}{10} \left (-5959+2665 \sqrt {5}\right )} \log \left (\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right ) \]
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Rubi [A]
time = 0.26, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {2094, 1687,
1692, 1183, 648, 632, 210, 642, 1677, 1674, 12} \begin {gather*} \frac {7}{4} \text {ArcTan}\left (\frac {1}{2} \left (\left (\frac {1}{x}+1\right )^2-1\right )\right )-\frac {1}{20} \sqrt {\frac {1}{10} \left (5959+2665 \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}{20} \sqrt {\frac {1}{10} \left (5959+2665 \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 {17-\left (\frac {1}{x}+1\right )^2}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {1}{40} \sqrt {\frac {1}{10} \left (2665 \sqrt {5}-5959\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}{40} \sqrt {\frac {1}{10} \left (2665 \sqrt {5}-5959\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 1183
Rule 1674
Rule 1677
Rule 1687
Rule 1692
Rule 2094
Rubi steps
\begin {align*} \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx &=-\left (16 \text {Subst}\left (\int \frac {(4-4 x)^6}{\left (1280-512 x^2+256 x^4\right )^2} \, dx,x,1+\frac {1}{x}\right )\right )\\ &=-\left (16 \text {Subst}\left (\int \frac {x \left (-24576-81920 x^2-24576 x^4\right )}{\left (1280-512 x^2+256 x^4\right )^2} \, dx,x,1+\frac {1}{x}\right )\right )-16 \text {Subst}\left (\int \frac {4096+61440 x^2+61440 x^4+4096 x^6}{\left (1280-512 x^2+256 x^4\right )^2} \, dx,x,1+\frac {1}{x}\right )\\ &=\frac {\left (59-17 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right )}{10 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}-\frac {\text {Subst}\left (\int \frac {261993005056+115964116992 x^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )}{167772160}-8 \text {Subst}\left (\int \frac {-24576-81920 x-24576 x^2}{\left (1280-512 x+256 x^2\right )^2} \, dx,x,\left (1+\frac {1}{x}\right )^2\right )\\ &=-\frac {17-\left (1+\frac {1}{x}\right )^2}{2 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {\left (59-17 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right )}{10 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}-\frac {\text {Subst}\left (\int -\frac {117440512}{1280-512 x+256 x^2} \, dx,x,\left (1+\frac {1}{x}\right )^2\right )}{131072}-\frac {\text {Subst}\left (\int \frac {261993005056 \sqrt {2 \left (1+\sqrt {5}\right )}-\left (261993005056-115964116992 \sqrt {5}\right ) x}{\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{85899345920 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\text {Subst}\left (\int \frac {261993005056 \sqrt {2 \left (1+\sqrt {5}\right )}+\left (261993005056-115964116992 \sqrt {5}\right ) x}{\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{85899345920 \sqrt {10 \left (1+\sqrt {5}\right )}}\\ &=-\frac {17-\left (1+\frac {1}{x}\right )^2}{2 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {\left (59-17 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right )}{10 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+896 \text {Subst}\left (\int \frac {1}{1280-512 x+256 x^2} \, dx,x,\left (1+\frac {1}{x}\right )^2\right )+\frac {\left (61-27 \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 )}{40 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (61-27 \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 )}{40 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {1}{20} \sqrt {\frac {1}{10} \left (3683+1647 \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} \sqrt {\frac {1}{10} \left (3683+1647 \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 {17-\left (1+\frac {1}{x}\right )^2}{2 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {\left (59-17 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right )}{10 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {1}{40} \sqrt {-\frac {5959}{10}+\frac {533 \sqrt {5}}{2}} \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}{40} \sqrt {-\frac {5959}{10}+\frac {533 \sqrt {5}}{2}} \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 )-1792 \text {Subst}\left (\int \frac {1}{-1048576-x^2} \, dx,x,-512+512 \left (1+\frac {1}{x}\right )^2\right )+\frac {1}{10} \sqrt {\frac {1}{10} \left (3683+1647 \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} \sqrt {\frac {1}{10} \left (3683+1647 \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 {17-\left (1+\frac {1}{x}\right )^2}{2 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {\left (59-17 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right )}{10 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {7}{4} \tan ^{-1}\left (\frac {1}{2} \left (-1+\left (1+\frac {1}{x}\right )^2\right )\right )-\frac {1}{20} \sqrt {\frac {1}{10} \left (5959+2665 \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}{20} \sqrt {\frac {1}{10} \left (5959+2665 \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}{40} \sqrt {-\frac {5959}{10}+\frac {533 \sqrt {5}}{2}} \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}{40} \sqrt {-\frac {5959}{10}+\frac {533 \sqrt {5}}{2}} \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 = 0.02, size = 108, normalized size = 0.34 \begin {gather*} \frac {1}{40} \left (\frac {38+84 x-32 x^2+72 x^3}{1+4 x+4 x^2+4 x^4}+\text {RootSum}\left [1+4 \text {$\#$1}+4 \text {$\#$1}^2+4 \text {$\#$1}^4\&,\frac {27 \log (x-\text {$\#$1})-16 \log (x-\text {$\#$1}) \text {$\#$1}+18 \log (x-\text {$\#$1}) \text {$\#$1}^2}{1+2 \text {$\#$1}+4 \text {$\#$1}^3}\&\right ]\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 = 79, normalized size = 0.25
method | result | size |
default | \(\frac {\frac {9}{20} x^{3}-\frac {1}{5} x^{2}+\frac {21}{40} x +\frac {19}{80}}{x^{4}+x^{2}+x +\frac {1}{4}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{\sum }\frac {\left (18 \textit {\_R}^{2}-16 \textit {\_R} +27\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+2 \textit {\_R} +1}\right )}{40}\) | \(79\) |
risch | \(\frac {\frac {9}{20} x^{3}-\frac {1}{5} x^{2}+\frac {21}{40} x +\frac {19}{80}}{x^{4}+x^{2}+x +\frac {1}{4}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{\sum }\frac {\left (18 \textit {\_R}^{2}-16 \textit {\_R} +27\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+2 \textit {\_R} +1}\right )}{40}\) | \(79\) |
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 = 1.16, size = 704, normalized size = 2.22 \begin {gather*} \text {Too large to display} \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. 3834 vs.
\(2 (257) = 514\).
time = 1.94, size = 3834, normalized size = 12.09 \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 = 5.62, size = 315, normalized size = 0.99 \begin {gather*} -\frac {1}{400} \, {\left (-\left (i + 3\right ) \, \sqrt {2665 \, \sqrt {5} - 4790} {\left (\frac {709 i}{533 \, \sqrt {5} - 958} + 1\right )} - 350 i\right )} \log \left (\left (2534636224790 i + 16853816172010\right ) \, \sqrt {5} x - \left (3913528401620 i + 26022625108780\right ) \, x + 5049076145 \, \sqrt {5} \sqrt {1424281 \, \sqrt {5} - 2199118} - \left (8426908086005 i - 1267318112395\right ) \, \sqrt {5} + \left (8166407345 i - 7795873310\right ) \, \sqrt {1424281 \, \sqrt {5} - 2199118} + 13011312554390 i - 1956764200810\right ) - \frac {1}{400} \, {\left (\left (i + 3\right ) \, \sqrt {2665 \, \sqrt {5} - 4790} {\left (\frac {709 i}{533 \, \sqrt {5} - 958} + 1\right )} - 350 i\right )} \log \left (\left (2534636224790 i + 16853816172010\right ) \, \sqrt {5} x - \left (3913528401620 i + 26022625108780\right ) \, x - 5049076145 \, \sqrt {5} \sqrt {1424281 \, \sqrt {5} - 2199118} - \left (8426908086005 i - 1267318112395\right ) \, \sqrt {5} - \left (8166407345 i - 7795873310\right ) \, \sqrt {1424281 \, \sqrt {5} - 2199118} + 13011312554390 i - 1956764200810\right ) - \frac {1}{400} \, {\left (\left (3 i + 1\right ) \, \sqrt {2665 \, \sqrt {5} + 4790} {\left (\frac {709 i}{533 \, \sqrt {5} + 958} + 1\right )} + 350 i\right )} \log \left (\left (16722951192450 i + 2480822188910\right ) \, \sqrt {5} x + \left (25712356272300 i + 3814385585140\right ) \, x + 5021907265 \, \sqrt {5} \sqrt {1416617 \, \sqrt {5} + 2178118} + \left (1240411094455 i - 8361475596225\right ) \, \sqrt {5} + \left (8153361745 i + 7721428310\right ) \, \sqrt {1416617 \, \sqrt {5} + 2178118} + 1907192792570 i - 12856178136150\right ) - \frac {1}{400} \, {\left (-\left (3 i + 1\right ) \, \sqrt {2665 \, \sqrt {5} + 4790} {\left (\frac {709 i}{533 \, \sqrt {5} + 958} + 1\right )} + 350 i\right )} \log \left (\left (16722951192450 i + 2480822188910\right ) \, \sqrt {5} x + \left (25712356272300 i + 3814385585140\right ) \, x - 5021907265 \, \sqrt {5} \sqrt {1416617 \, \sqrt {5} + 2178118} + \left (1240411094455 i - 8361475596225\right ) \, \sqrt {5} - \left (8153361745 i + 7721428310\right ) \, \sqrt {1416617 \, \sqrt {5} + 2178118} + 1907192792570 i - 12856178136150\right ) + \frac {36 \, x^{3} - 16 \, x^{2} + 42 \, x + 19}{20 \, {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.21, size = 174, normalized size = 0.55 \begin {gather*} \left (\sum _{k=1}^4\ln \left (-\frac {169\,\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}{100}+\frac {11\,x}{1600}+\frac {\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )\,x\,131}{100}-\frac {{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^2\,x\,72}{5}-{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^3\,x\,36+\frac {59\,{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^2}{20}-16\,{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^3+\frac {27}{1600}\right )\,\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )\right )+\frac {\frac {9\,x^3}{20}-\frac {x^2}{5}+\frac {21\,x}{40}+\frac {19}{80}}{x^4+x^2+x+\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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