Optimal. Leaf size=43 \[ -\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1-x^4\right )}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0332619, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1-x^4\right )}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[1/(x^8*(1 - 2*x^4 + x^8)),x]
[Out]
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Rubi in Sympy [A] time = 7.28232, size = 37, normalized size = 0.86 \[ \frac{11 \operatorname{atan}{\left (x \right )}}{8} + \frac{11 \operatorname{atanh}{\left (x \right )}}{8} - \frac{11}{12 x^{3}} - \frac{11}{28 x^{7}} + \frac{1}{4 x^{7} \left (- x^{4} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**8/(x**8-2*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0342513, size = 43, normalized size = 1. \[ \frac{1}{336} \left (-\frac{48}{x^7}-\frac{84 x}{x^4-1}-\frac{224}{x^3}-231 \log (1-x)+231 \log (x+1)+462 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^8*(1 - 2*x^4 + x^8)),x]
[Out]
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Maple [A] time = 0.024, size = 52, normalized size = 1.2 \[ -{\frac{1}{-16+16\,x}}-{\frac{11\,\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}+{\frac{11\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{7\,{x}^{7}}}-{\frac{2}{3\,{x}^{3}}}+{\frac{x}{8\,{x}^{2}+8}}+{\frac{11\,\arctan \left ( x \right ) }{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^8/(x^8-2*x^4+1),x)
[Out]
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Maxima [A] time = 0.854868, size = 57, normalized size = 1.33 \[ -\frac{77 \, x^{8} - 44 \, x^{4} - 12}{84 \,{\left (x^{11} - x^{7}\right )}} + \frac{11}{8} \, \arctan \left (x\right ) + \frac{11}{16} \, \log \left (x + 1\right ) - \frac{11}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - 2*x^4 + 1)*x^8),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280728, size = 92, normalized size = 2.14 \[ -\frac{308 \, x^{8} - 176 \, x^{4} - 462 \,{\left (x^{11} - x^{7}\right )} \arctan \left (x\right ) - 231 \,{\left (x^{11} - x^{7}\right )} \log \left (x + 1\right ) + 231 \,{\left (x^{11} - x^{7}\right )} \log \left (x - 1\right ) - 48}{336 \,{\left (x^{11} - x^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - 2*x^4 + 1)*x^8),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.730099, size = 44, normalized size = 1.02 \[ - \frac{11 \log{\left (x - 1 \right )}}{16} + \frac{11 \log{\left (x + 1 \right )}}{16} + \frac{11 \operatorname{atan}{\left (x \right )}}{8} - \frac{77 x^{8} - 44 x^{4} - 12}{84 x^{11} - 84 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**8/(x**8-2*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.280882, size = 55, normalized size = 1.28 \[ -\frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{14 \, x^{4} + 3}{21 \, x^{7}} + \frac{11}{8} \, \arctan \left (x\right ) + \frac{11}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{11}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 - 2*x^4 + 1)*x^8),x, algorithm="giac")
[Out]