Optimal. Leaf size=411 \[ -\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}} \]
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Rubi [A] time = 0.277631, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1374, 200, 31, 634, 617, 204, 628} \[ -\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 1374
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{1-x^3+x^6} \, dx &=-\left (\frac{1}{6} \left (-3+i \sqrt{3}\right ) \int \frac{1}{-\frac{1}{2}-\frac{i \sqrt{3}}{2}+x^3} \, dx\right )+\frac{1}{6} \left (3+i \sqrt{3}\right ) \int \frac{1}{-\frac{1}{2}+\frac{i \sqrt{3}}{2}+x^3} \, dx\\ &=\frac{\left (3-i \sqrt{3}\right ) \int \frac{1}{-\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \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}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \int \frac{1}{-\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \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}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}\\ &=\frac{\left (3+i \sqrt{3}\right ) \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \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}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \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}{6\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{\left (3+i \sqrt{3}\right ) \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}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \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}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}\\ &=\frac{\left (3+i \sqrt{3}\right ) \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \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 )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \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 )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \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 )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \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 )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}\\ &=-\frac{\left (i+\sqrt{3}\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (i-\sqrt{3}\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \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 )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \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 )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.008928, size = 39, normalized size = 0.09 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.004, size = 40, normalized size = 0.1 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{3}\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{x^{3}}{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 = 1.9112, size = 3903, normalized size = 9.5 \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 = 0.178717, size = 24, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (- 1458 t^{4} - 9 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15075, size = 860, normalized size = 2.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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