Optimal. Leaf size=186 \[ -\frac{(-1)^{5/18} \left (3 \log \left (\sqrt [9]{-1}-x\right )+\log (2)\right )}{9 \sqrt{3}}+\frac{(-1)^{13/18} \log \left (-\sqrt [3]{2} \left (x+(-1)^{8/9}\right )\right )}{3 \sqrt{3}}-\frac{(-1)^{13/18} \log \left (-2^{2/3} \left (\left ((-1)^{8/9}-x\right ) x+(-1)^{7/9}\right )\right )}{6 \sqrt{3}}+\frac{(-1)^{5/18} \log \left (2^{2/3} \left (x \left (x+\sqrt [9]{-1}\right )+(-1)^{2/9}\right )\right )}{6 \sqrt{3}}-\frac{1}{3} (-1)^{13/18} \tan ^{-1}\left (\frac{2 \sqrt [9]{-1} x+1}{\sqrt{3}}\right )+\frac{1}{3} (-1)^{5/18} \tan ^{-1}\left (\frac{1-2 (-1)^{8/9} x}{\sqrt{3}}\right ) \]
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Rubi [C] time = 0.24245, antiderivative size = 375, normalized size of antiderivative = 2.02, number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {1347, 200, 31, 634, 617, 204, 628} \[ -\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 1347
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{1-x^3+x^6} \, dx &=-\frac{i \int \frac{1}{-\frac{1}{2}-\frac{i \sqrt{3}}{2}+x^3} \, dx}{\sqrt{3}}+\frac{i \int \frac{1}{-\frac{1}{2}+\frac{i \sqrt{3}}{2}+x^3} \, dx}{\sqrt{3}}\\ &=\frac{i \int \frac{1}{-\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}+x} \, dx}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}+\frac{i \int \frac{-2^{2/3} \sqrt [3]{1-i \sqrt{3}}-x}{\left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}+\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )} x+x^2} \, dx}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \int \frac{1}{-\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}+x} \, dx}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \int \frac{-2^{2/3} \sqrt [3]{1+i \sqrt{3}}-x}{\left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}+\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )} x+x^2} \, dx}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}\\ &=\frac{i \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \int \frac{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}+2 x}{\left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}+\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )} x+x^2} \, dx}{3 \sqrt [3]{2} \sqrt{3} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{i \int \frac{1}{\left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}+\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )} x+x^2} \, dx}{2^{2/3} \sqrt{3} \sqrt [3]{1-i \sqrt{3}}}+\frac{i \int \frac{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}+2 x}{\left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}+\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )} x+x^2} \, dx}{3 \sqrt [3]{2} \sqrt{3} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{i \int \frac{1}{\left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}+\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )} x+x^2} \, dx}{2^{2/3} \sqrt{3} \sqrt [3]{1+i \sqrt{3}}}\\ &=\frac{i \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (\left (1-i \sqrt{3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (\left (1+i \sqrt{3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}\right )}{\sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}\right )}{\sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}\\ &=-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}+\frac{i \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (\left (1-i \sqrt{3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (\left (1+i \sqrt{3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1+i \sqrt{3}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0100208, size = 42, normalized size = 0.23 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.005, size = 37, normalized size = 0.2 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}} \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{1}{x^{6} - x^{3} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89363, size = 3900, normalized size = 20.97 \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 = 0.180478, size = 20, normalized size = 0.11 \begin{align*} \operatorname{RootSum}{\left (19683 t^{6} - 243 t^{3} + 1, \left ( t \mapsto t \log{\left (729 t^{4} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17408, size = 849, normalized size = 4.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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