Optimal. Leaf size=451 \[ -\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]
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Rubi [A] time = 0.406547, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1420, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1420
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1+x^4}{1+3 x^4+x^8} \, dx &=\frac{1}{10} \left (5-\sqrt{5}\right ) \int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx+\frac{1}{10} \left (5+\sqrt{5}\right ) \int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx\\ &=\frac{\int \frac{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx}{2 \sqrt{10}}+\frac{\int \frac{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx}{2 \sqrt{10}}+\frac{\int \frac{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx}{2 \sqrt{10}}+\frac{\int \frac{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx}{2 \sqrt{10}}\\ &=\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}{4 \sqrt{5}}+\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}{4 \sqrt{5}}+\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}{4 \sqrt{5}}+\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}{4 \sqrt{5}}-\frac{\sqrt [4]{3+\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\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\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\ 2^{3/4} \sqrt{5}}-\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}{4 \sqrt{5} \sqrt [4]{2 \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}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}\\ &=-\frac{\sqrt [4]{3+\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\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\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\ 2^{3/4} \sqrt{5}}-\frac{\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{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\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{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}\\ &=-\frac{\tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}+\frac{\tan ^{-1}\left (1+\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\tan ^{-1}\left (1+\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\sqrt [4]{3+\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\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\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\ 2^{3/4} \sqrt{5}}-\frac{\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{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\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{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}\\ \end{align*}
Mathematica [C] time = 0.0141974, size = 55, normalized size = 0.12 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 42, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ({{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} + 1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62415, size = 3098, normalized size = 6.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.06069, size = 24, normalized size = 0.05 \begin{align*} \operatorname{RootSum}{\left (40960000 t^{8} + 19200 t^{4} + 1, \left ( t \mapsto t \log{\left (25600 t^{5} + 16 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35736, size = 342, normalized size = 0.76 \begin{align*} \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x + 130 \, i \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x - 130 \, i \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x + 130 \, \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x - 130 \, \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x + 50 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x - 50 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x + 50 \, \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x - 50 \, \sqrt{\sqrt{5} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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