Optimal. Leaf size=382 \[ -\frac{x^2}{2}-\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1-i \sqrt{3}}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1+i \sqrt{3}}}+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}} \]
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Rubi [A] time = 0.777977, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ -\frac{x^2}{2}-\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1-i \sqrt{3}}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1+i \sqrt{3}}}+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(1 - x^3))/(1 - x^3 + x^6),x]
[Out]
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Rubi in Sympy [A] time = 125.338, size = 340, normalized size = 0.89 \[ - \frac{x^{2}}{2} + \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{9 \sqrt [3]{1 - \sqrt{3} i}} - \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{9 \sqrt [3]{1 + \sqrt{3} i}} - \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{18 \sqrt [3]{1 - \sqrt{3} i}} + \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{18 \sqrt [3]{1 + \sqrt{3} i}} + \frac{\sqrt [3]{2} i \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{1 - \sqrt{3} i}} - \frac{\sqrt [3]{2} i \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{1 + \sqrt{3} i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(-x**3+1)/(x**6-x**3+1),x)
[Out]
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Mathematica [C] time = 0.0202578, size = 48, normalized size = 0.13 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-\text{$\#$1}}\&\right ]-\frac{x^2}{2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(1 - x^3))/(1 - x^3 + x^6),x]
[Out]
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Maple [C] time = 0.008, size = 44, normalized size = 0.1 \[ -{\frac{{x}^{2}}{2}}+{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(-x^3+1)/(x^6-x^3+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2} \, x^{2} + \int \frac{x}{x^{6} - x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 1)*x^4/(x^6 - x^3 + 1),x, algorithm="maxima")
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Fricas [A] time = 0.282208, size = 1447, normalized size = 3.79 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 1)*x^4/(x^6 - x^3 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.492278, size = 32, normalized size = 0.08 \[ - \frac{x^{2}}{2} - \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (- 6561 t^{5} - 27 t^{2} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(-x**3+1)/(x**6-x**3+1),x)
[Out]
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GIAC/XCAS [A] time = 0.286148, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 1)*x^4/(x^6 - x^3 + 1),x, algorithm="giac")
[Out]