Optimal. Leaf size=378 \[ -\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1+i \sqrt{3}\right )^{2/3}}-x+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}} \]
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Rubi [A] time = 0.25865, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {1502, 1347, 200, 31, 634, 617, 204, 628} \[ -\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1+i \sqrt{3}\right )^{2/3}}-x+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 1502
Rule 1347
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 \left (1-x^3\right )}{1-x^3+x^6} \, dx &=-x+\int \frac{1}{1-x^3+x^6} \, dx\\ &=-x-\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}}\\ &=-x+\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}}\\ &=-x+\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}}}\\ &=-x+\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}}\\ &=-x-\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 = 0.0127722, size = 46, normalized size = 0.12 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\& \right ]-x \]
Antiderivative was successfully verified.
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Maple [C] time = 0.004, size = 41, normalized size = 0.1 \begin{align*} -x+{\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*} -x + \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 = 1.67323, size = 3906, normalized size = 10.33 \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 = 0.17863, size = 24, normalized size = 0.06 \begin{align*} - x - \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 = 1.17756, size = 853, normalized size = 2.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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