Optimal. Leaf size=355 \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]
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Rubi [A] time = 0.277298, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1421, 1169, 634, 618, 204, 628} \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]
Antiderivative was successfully verified.
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Rule 1421
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1-x^4}{1-x^4+x^8} \, dx &=-\frac{\int \frac{\sqrt{3}+2 x^2}{-1-\sqrt{3} x^2-x^4} \, dx}{2 \sqrt{3}}-\frac{\int \frac{\sqrt{3}-2 x^2}{-1+\sqrt{3} x^2-x^4} \, dx}{2 \sqrt{3}}\\ &=\frac{\int \frac{\sqrt{3 \left (2-\sqrt{3}\right )}-\left (-2+\sqrt{3}\right ) x}{1-\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\int \frac{\sqrt{3 \left (2-\sqrt{3}\right )}+\left (-2+\sqrt{3}\right ) x}{1+\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\int \frac{\sqrt{3 \left (2+\sqrt{3}\right )}-\left (2+\sqrt{3}\right ) x}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\int \frac{\sqrt{3 \left (2+\sqrt{3}\right )}+\left (2+\sqrt{3}\right ) x}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}\\ &=-\left (\frac{1}{8} \sqrt{\frac{1}{3} \left (7-4 \sqrt{3}\right )} \int \frac{1}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (7-4 \sqrt{3}\right )} \int \frac{1}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \int \frac{-\sqrt{2-\sqrt{3}}+2 x}{1-\sqrt{2-\sqrt{3}} x+x^2} \, dx+\frac{\left (-2+\sqrt{3}\right ) \int \frac{\sqrt{2-\sqrt{3}}+2 x}{1+\sqrt{2-\sqrt{3}} x+x^2} \, dx}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \int \frac{-\sqrt{2+\sqrt{3}}+2 x}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \int \frac{\sqrt{2+\sqrt{3}}+2 x}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx+\frac{1}{8} \sqrt{\frac{1}{3} \left (7+4 \sqrt{3}\right )} \int \frac{1}{1-\sqrt{2-\sqrt{3}} x+x^2} \, dx+\frac{1}{8} \sqrt{\frac{1}{3} \left (7+4 \sqrt{3}\right )} \int \frac{1}{1+\sqrt{2-\sqrt{3}} x+x^2} \, dx\\ &=\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )-\frac{1}{8} \sqrt{\frac{2}{3}-\frac{1}{\sqrt{3}}} \log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (1-\sqrt{2+\sqrt{3}} x+x^2\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (1+\sqrt{2+\sqrt{3}} x+x^2\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (7-4 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,-\sqrt{2+\sqrt{3}}+2 x\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (7-4 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,\sqrt{2+\sqrt{3}}+2 x\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (7+4 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{3}-x^2} \, dx,x,-\sqrt{2-\sqrt{3}}+2 x\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (7+4 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{3}-x^2} \, dx,x,\sqrt{2-\sqrt{3}}+2 x\right )\\ &=-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}+2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}+2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )-\frac{1}{8} \sqrt{\frac{2}{3}-\frac{1}{\sqrt{3}}} \log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (1-\sqrt{2+\sqrt{3}} x+x^2\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (1+\sqrt{2+\sqrt{3}} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.0153069, size = 57, normalized size = 0.16 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 44, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\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} - x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.51969, size = 2313, normalized size = 6.52 \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 = 1.25847, size = 26, normalized size = 0.07 \begin{align*} - \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (9216 t^{5} - 8 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17265, size = 342, normalized size = 0.96 \begin{align*} \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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