Optimal. Leaf size=411 \[ -\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}} \]
[Out]
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Rubi [A] time = 0.714557, 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 \[ -\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(1 - x^3 + x^6),x]
[Out]
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Rubi in Sympy [A] time = 99.8014, size = 335, normalized size = 0.82 \[ \frac{\sqrt [3]{2} \sqrt{3} i \left (1 - \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{18} - \frac{\sqrt [3]{2} \sqrt{3} i \left (1 + \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{18} - \frac{\sqrt [3]{2} \sqrt{3} i \left (1 - \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} + \frac{\sqrt [3]{2} \sqrt{3} i \left (1 + \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} + \frac{\sqrt [3]{2} i \left (1 - \sqrt{3} i\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt [3]{2} i \left (1 + \sqrt{3} i\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(x**6-x**3+1),x)
[Out]
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Mathematica [C] time = 0.0158241, size = 41, normalized size = 0.1 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(1 - x^3 + x^6),x]
[Out]
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Maple [C] time = 0.006, size = 40, normalized size = 0.1 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(x^6-x^3+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{x^{6} - x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^6 - x^3 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271635, size = 1436, normalized size = 3.49 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^6 - x^3 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.473822, size = 26, normalized size = 0.06 \[ \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (6561 t^{5} + 54 t^{2} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(x**6-x**3+1),x)
[Out]
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GIAC/XCAS [A] time = 0.276396, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^6 - x^3 + 1),x, algorithm="giac")
[Out]