Optimal. Leaf size=416 \[ -\frac {1}{x}+\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )+\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )-\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )-\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1382, 1524,
303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\left (3+\sqrt {5}\right )^{5/4} \text {ArcTan}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \text {ArcTan}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \text {ArcTan}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )+\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \text {ArcTan}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )-\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )-\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1382
Rule 1524
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx &=-\frac {1}{x}+\int \frac {x^2 \left (-3-x^4\right )}{1+3 x^4+x^8} \, dx\\ &=-\frac {1}{x}+\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=-\frac {1}{x}-\frac {\left (3-\sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}+\frac {\left (3-\sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right ) \int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right ) \int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{4 \sqrt {10}}\\ &=-\frac {1}{x}-\frac {\left (3+\sqrt {5}\right )^{5/4} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{8\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3-\sqrt {5}\right ) \int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{8 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\left (3-\sqrt {5}\right ) \int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{8 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {1}{40} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx+\frac {1}{40} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx\\ &=-\frac {1}{x}-\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )-\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )+-\frac {\left (-5-3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{20 \sqrt [4]{2 \left (3-\sqrt {5}\right )}}+\frac {\left (5-3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{20 \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\left (-5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{20 \sqrt [4]{2 \left (3+\sqrt {5}\right )}}-\frac {\left (5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{20 \sqrt [4]{2 \left (3-\sqrt {5}\right )}}\\ &=-\frac {1}{x}+\frac {\sqrt [4]{246+110 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{246+110 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt {5}}-\frac {1}{20} \sqrt [4]{6150-2750 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )+\frac {\sqrt [4]{246-110 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )-\frac {1}{40} \sqrt [4]{6150-2750 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^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 = 61, normalized size = 0.15 \begin {gather*} -\frac {1}{x}-\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}+2 \text {$\#$1}^5}\&\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 = 0.03, size = 52, normalized size = 0.12
method | result | size |
risch | \(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (625 \textit {\_Z}^{8}+3075 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (1175 \textit {\_R}^{7}+5778 \textit {\_R}^{3}+11 x \right )\right )}{4}\) | \(42\) |
default | \(-\frac {1}{x}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(52\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1025 vs.
\(2 (270) = 540\).
time = 0.39, size = 1025, normalized size = 2.46 \begin {gather*} -\frac {\sqrt {10} {\left (55 \, \sqrt {5} x - 123 \, x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \sqrt {55 \, \sqrt {5} + 123} \arctan \left (\frac {1}{800} \, \sqrt {10} \sqrt {-40 \, \sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x - 105 \, \sqrt {2} x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 1600 \, x^{2} - 800 \, \sqrt {110 \, \sqrt {5} + 246} {\left (4 \, \sqrt {5} - 9\right )}} {\left (161 \, \sqrt {5} - 360\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \sqrt {55 \, \sqrt {5} + 123} - \frac {1}{20} \, \sqrt {10} {\left (161 \, \sqrt {5} x - 360 \, x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \sqrt {55 \, \sqrt {5} + 123} - \frac {1}{8} \, {\left (55 \, \sqrt {5} \sqrt {2} - 123 \, \sqrt {2}\right )} \sqrt {110 \, \sqrt {5} + 246} \sqrt {55 \, \sqrt {5} + 123}\right ) + \sqrt {10} {\left (55 \, \sqrt {5} x + 123 \, x\right )} \sqrt {-55 \, \sqrt {5} + 123} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \arctan \left (\frac {1}{800} \, \sqrt {10} \sqrt {-40 \, \sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x + 105 \, \sqrt {2} x\right )} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 1600 \, x^{2} + 800 \, {\left (4 \, \sqrt {5} + 9\right )} \sqrt {-110 \, \sqrt {5} + 246}} {\left (161 \, \sqrt {5} + 360\right )} \sqrt {-55 \, \sqrt {5} + 123} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} - \frac {1}{40} \, {\left (2 \, \sqrt {10} {\left (161 \, \sqrt {5} x + 360 \, x\right )} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} - 5 \, {\left (55 \, \sqrt {5} \sqrt {2} + 123 \, \sqrt {2}\right )} \sqrt {-110 \, \sqrt {5} + 246}\right )} \sqrt {-55 \, \sqrt {5} + 123}\right ) + \sqrt {10} {\left (55 \, \sqrt {5} x - 123 \, x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \sqrt {55 \, \sqrt {5} + 123} \arctan \left (-\frac {1}{20} \, \sqrt {10} {\left (161 \, \sqrt {5} x - 360 \, x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \sqrt {55 \, \sqrt {5} + 123} + \frac {1}{40} \, \sqrt {\sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x - 105 \, \sqrt {2} x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 40 \, x^{2} - 20 \, \sqrt {110 \, \sqrt {5} + 246} {\left (4 \, \sqrt {5} - 9\right )}} {\left (161 \, \sqrt {5} - 360\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \sqrt {55 \, \sqrt {5} + 123} + \frac {1}{8} \, {\left (55 \, \sqrt {5} \sqrt {2} - 123 \, \sqrt {2}\right )} \sqrt {110 \, \sqrt {5} + 246} \sqrt {55 \, \sqrt {5} + 123}\right ) + \sqrt {10} {\left (55 \, \sqrt {5} x + 123 \, x\right )} \sqrt {-55 \, \sqrt {5} + 123} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} \arctan \left (\frac {1}{40} \, \sqrt {\sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x + 105 \, \sqrt {2} x\right )} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 40 \, x^{2} + 20 \, {\left (4 \, \sqrt {5} + 9\right )} \sqrt {-110 \, \sqrt {5} + 246}} {\left (161 \, \sqrt {5} + 360\right )} \sqrt {-55 \, \sqrt {5} + 123} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} - \frac {1}{40} \, {\left (2 \, \sqrt {10} {\left (161 \, \sqrt {5} x + 360 \, x\right )} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 5 \, {\left (55 \, \sqrt {5} \sqrt {2} + 123 \, \sqrt {2}\right )} \sqrt {-110 \, \sqrt {5} + 246}\right )} \sqrt {-55 \, \sqrt {5} + 123}\right ) - \sqrt {10} \sqrt {2} x {\left (110 \, \sqrt {5} + 246\right )}^{\frac {1}{4}} \log \left (40 \, \sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x - 105 \, \sqrt {2} x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 1600 \, x^{2} - 800 \, \sqrt {110 \, \sqrt {5} + 246} {\left (4 \, \sqrt {5} - 9\right )}\right ) + \sqrt {10} \sqrt {2} x {\left (110 \, \sqrt {5} + 246\right )}^{\frac {1}{4}} \log \left (-40 \, \sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x - 105 \, \sqrt {2} x\right )} {\left (110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 1600 \, x^{2} - 800 \, \sqrt {110 \, \sqrt {5} + 246} {\left (4 \, \sqrt {5} - 9\right )}\right ) + \sqrt {10} \sqrt {2} x {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {1}{4}} \log \left (40 \, \sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x + 105 \, \sqrt {2} x\right )} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 1600 \, x^{2} + 800 \, {\left (4 \, \sqrt {5} + 9\right )} \sqrt {-110 \, \sqrt {5} + 246}\right ) - \sqrt {10} \sqrt {2} x {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {1}{4}} \log \left (-40 \, \sqrt {10} {\left (47 \, \sqrt {5} \sqrt {2} x + 105 \, \sqrt {2} x\right )} {\left (-110 \, \sqrt {5} + 246\right )}^{\frac {3}{4}} + 1600 \, x^{2} + 800 \, {\left (4 \, \sqrt {5} + 9\right )} \sqrt {-110 \, \sqrt {5} + 246}\right ) + 80}{80 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.14, size = 32, normalized size = 0.08 \begin {gather*} \operatorname {RootSum} {\left (40960000 t^{8} + 787200 t^{4} + 1, \left ( t \mapsto t \log {\left (\frac {19251200 t^{7}}{11} + \frac {369792 t^{3}}{11} + x \right )} \right )\right )} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.68, size = 244, normalized size = 0.59 \begin {gather*} -\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (748225 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 748225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (748225 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 748225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (180625 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 180625 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (180625 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 180625 \, x^{2}\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.29, size = 292, normalized size = 0.70 \begin {gather*} -\frac {1}{x}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2585\,2^{3/4}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {1155\,2^{3/4}\,\sqrt {5}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}+6765\right )}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2585\,2^{3/4}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {1155\,2^{3/4}\,\sqrt {5}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,2585{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,2585{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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