Optimal. Leaf size=170 \[ x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1367, 1422, 212, 206, 203} \[ x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 212
Rule 1367
Rule 1422
Rubi steps
\begin {align*} \int \frac {x^8}{1-3 x^4+x^8} \, dx &=x-\int \frac {1-3 x^4}{1-3 x^4+x^8} \, dx\\ &=x-\frac {1}{10} \left (-15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=x+\sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx+\sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx--\frac {\left (-15-7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}--\frac {\left (-15-7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}\\ &=x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 160, normalized size = 0.94 \[ x+\frac {\left (\sqrt {5}-2\right ) \tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {10 \left (\sqrt {5}-1\right )}}-\frac {\left (2+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\left (\sqrt {5}-2\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {10 \left (\sqrt {5}-1\right )}}-\frac {\left (2+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.87, size = 304, normalized size = 1.79 \[ -\frac {1}{10} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \arctan \left (\frac {1}{20} \, {\left (\sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} + 1} {\left (2 \, \sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} - 2 \, \sqrt {10} {\left (2 \, \sqrt {5} x - 5 \, x\right )}\right )} \sqrt {5 \, \sqrt {5} + 11}\right ) - \frac {1}{10} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \arctan \left (\frac {1}{20} \, {\left (\sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (2 \, \sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} - 2 \, \sqrt {10} {\left (2 \, \sqrt {5} x + 5 \, x\right )}\right )} \sqrt {5 \, \sqrt {5} - 11}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.68, size = 148, normalized size = 0.87 \[ -\frac {1}{20} \, \sqrt {50 \, \sqrt {5} + 110} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {50 \, \sqrt {5} - 110} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {50 \, \sqrt {5} - 110} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {50 \, \sqrt {5} - 110} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 205, normalized size = 1.21 \[ x +\frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}-\frac {2 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-\frac {2 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {\arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}-\frac {2 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-\frac {2 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x + \frac {1}{2} \, \int \frac {2 \, x^{2} + 1}{x^{4} - x^{2} - 1}\,{d x} - \frac {1}{2} \, \int \frac {2 \, x^{2} - 1}{x^{4} + x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.44, size = 246, normalized size = 1.45 \[ x-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-50\,\sqrt {5}-110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}+\frac {\sqrt {5}\,x\,\sqrt {-50\,\sqrt {5}-110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}\right )\,\sqrt {-50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {110-50\,\sqrt {5}}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}-\frac {\sqrt {5}\,x\,\sqrt {110-50\,\sqrt {5}}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}\right )\,\sqrt {110-50\,\sqrt {5}}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}-110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}-\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}-110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}\right )\,\sqrt {50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}+110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}+\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}+110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}\right )\,\sqrt {50\,\sqrt {5}+110}\,1{}\mathrm {i}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.24, size = 58, normalized size = 0.34 \[ x + \operatorname {RootSum} {\left (6400 t^{4} - 880 t^{2} - 1, \left (t \mapsto t \log {\left (- \frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 880 t^{2} - 1, \left (t \mapsto t \log {\left (- \frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________