Optimal. Leaf size=100 \[ -\frac{1}{2 x^2}+\frac{\tan ^{-1}\left (1-\sqrt{2} x^2\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x^2+1\right )}{4 \sqrt{2}}-\frac{\log \left (x^4-\sqrt{2} x^2+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^4+\sqrt{2} x^2+1\right )}{8 \sqrt{2}} \]
[Out]
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Rubi [A] time = 0.153243, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ -\frac{1}{2 x^2}+\frac{\tan ^{-1}\left (1-\sqrt{2} x^2\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x^2+1\right )}{4 \sqrt{2}}-\frac{\log \left (x^4-\sqrt{2} x^2+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^4+\sqrt{2} x^2+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(1 + x^8)),x]
[Out]
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Rubi in Sympy [A] time = 18.0573, size = 87, normalized size = 0.87 \[ - \frac{\sqrt{2} \log{\left (x^{4} - \sqrt{2} x^{2} + 1 \right )}}{16} + \frac{\sqrt{2} \log{\left (x^{4} + \sqrt{2} x^{2} + 1 \right )}}{16} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} - 1 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} + 1 \right )}}{8} - \frac{1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(x**8+1),x)
[Out]
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Mathematica [B] time = 0.0879892, size = 208, normalized size = 2.08 \[ -\frac{1}{2 x^2}+\frac{\log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )}{8 \sqrt{2}}-\frac{\tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(1 + x^8)),x]
[Out]
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Maple [A] time = 0.006, size = 71, normalized size = 0.7 \[ -{\frac{\arctan \left ( 1+{x}^{2}\sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\arctan \left ({x}^{2}\sqrt{2}-1 \right ) \sqrt{2}}{8}}-{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{4}-{x}^{2}\sqrt{2}}{1+{x}^{4}+{x}^{2}\sqrt{2}}} \right ) }-{\frac{1}{2\,{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(x^8+1),x)
[Out]
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Maxima [A] time = 1.58442, size = 115, normalized size = 1.15 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} - \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{4} + \sqrt{2} x^{2} + 1\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{4} - \sqrt{2} x^{2} + 1\right ) - \frac{1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 1)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231046, size = 170, normalized size = 1.7 \[ \frac{4 \, \sqrt{2} x^{2} \arctan \left (\frac{1}{\sqrt{2} x^{2} + \sqrt{2} \sqrt{x^{4} + \sqrt{2} x^{2} + 1} + 1}\right ) + 4 \, \sqrt{2} x^{2} \arctan \left (\frac{1}{\sqrt{2} x^{2} + \sqrt{2} \sqrt{x^{4} - \sqrt{2} x^{2} + 1} - 1}\right ) + \sqrt{2} x^{2} \log \left (x^{4} + \sqrt{2} x^{2} + 1\right ) - \sqrt{2} x^{2} \log \left (x^{4} - \sqrt{2} x^{2} + 1\right ) - 8}{16 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 1)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.561488, size = 87, normalized size = 0.87 \[ - \frac{\sqrt{2} \log{\left (x^{4} - \sqrt{2} x^{2} + 1 \right )}}{16} + \frac{\sqrt{2} \log{\left (x^{4} + \sqrt{2} x^{2} + 1 \right )}}{16} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} - 1 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} + 1 \right )}}{8} - \frac{1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(x**8+1),x)
[Out]
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GIAC/XCAS [A] time = 0.233584, size = 131, normalized size = 1.31 \[ -\frac{1}{8} \, \sqrt{2} x^{4} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} x^{4} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2} x^{4}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} + 1\right ) + \frac{1}{16} \, \sqrt{2} x^{4}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} + 1\right ) - \frac{1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 1)*x^3),x, algorithm="giac")
[Out]