Optimal. Leaf size=414 \[ -\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2 \sqrt {10}}+\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {1347, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2 \sqrt {10}}+\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1347
Rubi steps
\begin {align*} \int \frac {1}{1+3 x^4+x^8} \, dx &=\frac {\int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}-\frac {\int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}\\ &=\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3-\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3-\sqrt {5}\right )}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3+\sqrt {5}\right )}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3+\sqrt {5}\right )}}\\ &=\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx}{2 \sqrt {10 \left (3-\sqrt {5}\right )}}+\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx}{2 \sqrt {10 \left (3-\sqrt {5}\right )}}+\frac {\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}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}+\frac {\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}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx}{2 \sqrt {10 \left (3+\sqrt {5}\right )}}-\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx}{2 \sqrt {10 \left (3+\sqrt {5}\right )}}-\frac {\sqrt [4]{9+4 \sqrt {5}} \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}{4 \sqrt {10}}-\frac {\sqrt [4]{9+4 \sqrt {5}} \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}{4 \sqrt {10}}\\ &=-\frac {\sqrt [4]{9+4 \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 )}{4 \sqrt {10}}+\frac {\sqrt [4]{9+4 \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 )}{4 \sqrt {10}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}\\ &=-\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}+\frac {\tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}+\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\sqrt [4]{9+4 \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 )}{4 \sqrt {10}}+\frac {\sqrt [4]{9+4 \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 )}{4 \sqrt {10}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 42, normalized size = 0.10 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^7+3 \text {$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.99, size = 733, normalized size = 1.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.53, size = 239, normalized size = 0.58 \[ \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.01, size = 37, normalized size = 0.09 \[ \frac {\ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+12 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 403, normalized size = 0.97 \[ \frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {64\,\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {64\,\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.52, size = 26, normalized size = 0.06 \[ \operatorname {RootSum} {\left (40960000 t^{8} + 115200 t^{4} + 1, \left (t \mapsto t \log {\left (- 9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________