Optimal. Leaf size=268 \[ -\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2-\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )+\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2+\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )-\frac{\tan ^{-1}\left (\frac{3-\left (\frac{4}{x}+1\right )^2}{6 \sqrt{7}}\right )}{12 \sqrt{7}}-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right )-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right ) \]
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Rubi [A] time = 0.868381, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ -\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2-\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )+\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2+\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )-\frac{\tan ^{-1}\left (\frac{3-\left (\frac{4}{x}+1\right )^2}{6 \sqrt{7}}\right )}{12 \sqrt{7}}-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right )-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(8 + 8*x - x^3 + 8*x^4)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 65.2582, size = 382, normalized size = 1.43 \[ \frac{\sqrt{174} \left (- 96 \sqrt{29} + 32\right ) \log{\left (\sqrt{6} \left (- \frac{1}{16} - \frac{1}{4 x}\right ) \sqrt{1 + \sqrt{29}} + \left (\frac{1}{4} + \frac{1}{x}\right )^{2} + \frac{3 \sqrt{29}}{16} \right )}}{133632 \sqrt{1 + \sqrt{29}}} - \frac{\sqrt{174} \left (- 96 \sqrt{29} + 32\right ) \log{\left (\sqrt{6} \left (\frac{1}{16} + \frac{1}{4 x}\right ) \sqrt{1 + \sqrt{29}} + \left (\frac{1}{4} + \frac{1}{x}\right )^{2} + \frac{3 \sqrt{29}}{16} \right )}}{133632 \sqrt{1 + \sqrt{29}}} + \frac{\sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{8 \left (\frac{1}{4} + \frac{1}{x}\right )^{2}}{21} - \frac{1}{14}\right ) \right )}}{84} - \frac{\sqrt{29} \left (16 \sqrt{6} \sqrt{1 + \sqrt{29}} - \frac{\sqrt{6} \sqrt{1 + \sqrt{29}} \left (- 192 \sqrt{29} + 64\right )}{8}\right ) \operatorname{atan}{\left (\frac{\sqrt{6} \left (\frac{1}{3} + \frac{\sqrt{6 + 6 \sqrt{29}}}{6} + \frac{4}{3 x}\right )}{\sqrt{-1 + \sqrt{29}}} \right )}}{16704 \sqrt{-1 + \sqrt{29}} \sqrt{1 + \sqrt{29}}} - \frac{\sqrt{29} \left (16 \sqrt{6} \sqrt{1 + \sqrt{29}} - \frac{\sqrt{6} \sqrt{1 + \sqrt{29}} \left (- 192 \sqrt{29} + 64\right )}{8}\right ) \operatorname{atan}{\left (\frac{\sqrt{6} \left (- \frac{\sqrt{6 + 6 \sqrt{29}}}{6} + \frac{1}{3} + \frac{4}{3 x}\right )}{\sqrt{-1 + \sqrt{29}}} \right )}}{16704 \sqrt{-1 + \sqrt{29}} \sqrt{1 + \sqrt{29}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(8*x**4-x**3+8*x+8),x)
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Mathematica [C] time = 0.0137583, size = 45, normalized size = 0.17 \[ \text{RootSum}\left [8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,\frac{\log (x-\text{$\#$1})}{32 \text{$\#$1}^3-3 \text{$\#$1}^2+8}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[(8 + 8*x - x^3 + 8*x^4)^(-1),x]
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Maple [C] time = 0.007, size = 41, normalized size = 0.2 \[ \sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}-{{\it \_Z}}^{3}+8\,{\it \_Z}+8 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(8*x^4-x^3+8*x+8),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{8 \, x^{4} - x^{3} + 8 \, x + 8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(8*x^4 - x^3 + 8*x + 8),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(8*x^4 - x^3 + 8*x + 8),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.04107, size = 41, normalized size = 0.15 \[ \operatorname{RootSum}{\left (66298176 t^{4} + 74088 t^{2} + 4095 t + 64, \left ( t \mapsto t \log{\left (\frac{35914274424 t^{3}}{2109763} - \frac{1504863360 t^{2}}{2109763} + \frac{102851343 t}{2109763} + x + \frac{6055613}{16878104} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(8*x**4-x**3+8*x+8),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{8 \, x^{4} - x^{3} + 8 \, x + 8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(8*x^4 - x^3 + 8*x + 8),x, algorithm="giac")
[Out]