Optimal. Leaf size=418 \[ -\frac{1}{2 x^2}+\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}} \]
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Rubi [A] time = 0.854118, antiderivative size = 418, 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 \[ -\frac{1}{2 x^2}+\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[(1 - x^3)/(x^3*(1 - x^3 + x^6)),x]
[Out]
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Rubi in Sympy [A] time = 129.123, size = 342, normalized size = 0.82 \[ - \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 - \sqrt{3} i} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{18} + \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 + \sqrt{3} i} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{18} + \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 - \sqrt{3} i} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} - \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 + \sqrt{3} i} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} + \frac{2^{\frac{2}{3}} i \sqrt [3]{1 - \sqrt{3} i} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} - \frac{2^{\frac{2}{3}} i \sqrt [3]{1 + \sqrt{3} i} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} - \frac{1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**3+1)/x**3/(x**6-x**3+1),x)
[Out]
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Mathematica [C] time = 0.0204511, size = 47, normalized size = 0.11 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right ]-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^3)/(x^3*(1 - x^3 + x^6)),x]
[Out]
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Maple [C] time = 0.009, size = 46, normalized size = 0.1 \[ -{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{3}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}}-{\frac{1}{2\,{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^3+1)/x^3/(x^6-x^3+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2 \, x^{2}} - \int \frac{x^{3}}{x^{6} - x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^3),x, algorithm="maxima")
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Fricas [A] time = 0.279619, size = 930, normalized size = 2.22 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.562082, size = 32, normalized size = 0.08 \[ - \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (- 1458 t^{4} - 9 t + x \right )} \right )\right )} - \frac{1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**3+1)/x**3/(x**6-x**3+1),x)
[Out]
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GIAC/XCAS [A] time = 0.297666, size = 867, normalized size = 2.07 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^3),x, algorithm="giac")
[Out]