Optimal. Leaf size=91 \[ \frac{x}{32 \left (1-x^2\right )}+\frac{\left (99-17 x^2\right ) x}{128 \left (x^4-6 x^2+1\right )}+\frac{5}{32} \tanh ^{-1}(x)+\frac{1}{512} \left (3 \sqrt{2}-4\right ) \tanh ^{-1}\left (\left (\sqrt{2}-1\right ) x\right )+\frac{1}{512} \left (4+3 \sqrt{2}\right ) \tanh ^{-1}\left (\left (1+\sqrt{2}\right ) x\right ) \]
[Out]
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Rubi [B] time = 0.264653, antiderivative size = 205, normalized size of antiderivative = 2.25, number of steps used = 15, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{41-17 x}{256 \left (-x^2+2 x+1\right )}+\frac{17 x+41}{256 \left (-x^2-2 x+1\right )}+\frac{1}{64 (1-x)}-\frac{1}{64 (x+1)}+\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (-x-\sqrt{2}+1\right )+\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (-x+\sqrt{2}+1\right )-\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )-\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right )-\frac{17 \tanh ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{256 \sqrt{2}}+\frac{5}{32} \tanh ^{-1}(x)+\frac{17 \tanh ^{-1}\left (\frac{x+1}{\sqrt{2}}\right )}{256 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(-1 + 7*x^2 - 7*x^4 + x^6)^(-2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**6-7*x**4+7*x**2-1)**2,x)
[Out]
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Mathematica [A] time = 0.146309, size = 132, normalized size = 1.45 \[ \frac{-\frac{8 x \left (21 x^4-140 x^2+103\right )}{x^6-7 x^4+7 x^2-1}-80 \log (1-x)-\left (4+3 \sqrt{2}\right ) \log \left (-x+\sqrt{2}-1\right )+\left (4-3 \sqrt{2}\right ) \log \left (-x+\sqrt{2}+1\right )+80 \log (x+1)+\left (4+3 \sqrt{2}\right ) \log \left (x+\sqrt{2}-1\right )+\left (3 \sqrt{2}-4\right ) \log \left (x+\sqrt{2}+1\right )}{1024} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + 7*x^2 - 7*x^4 + x^6)^(-2),x]
[Out]
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Maple [A] time = 0.028, size = 116, normalized size = 1.3 \[{\frac{1}{128\,{x}^{2}-256\,x-128} \left ( -{\frac{17\,x}{2}}+{\frac{41}{2}} \right ) }+{\frac{\ln \left ({x}^{2}-2\,x-1 \right ) }{256}}+{\frac{3\,\sqrt{2}}{512}{\it Artanh} \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{1}{-64+64\,x}}-{\frac{5\,\ln \left ( -1+x \right ) }{64}}-{\frac{1}{64+64\,x}}+{\frac{5\,\ln \left ( 1+x \right ) }{64}}-{\frac{1}{128\,{x}^{2}+256\,x-128} \left ({\frac{17\,x}{2}}+{\frac{41}{2}} \right ) }-{\frac{\ln \left ({x}^{2}+2\,x-1 \right ) }{256}}+{\frac{3\,\sqrt{2}}{512}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{2}}{4}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^6-7*x^4+7*x^2-1)^2,x)
[Out]
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Maxima [A] time = 0.888377, size = 162, normalized size = 1.78 \[ -\frac{3}{1024} \, \sqrt{2} \log \left (\frac{2 \,{\left (x - \sqrt{2} + 1\right )}}{2 \, x + 2 \, \sqrt{2} + 2}\right ) - \frac{3}{1024} \, \sqrt{2} \log \left (\frac{2 \,{\left (x - \sqrt{2} - 1\right )}}{2 \, x + 2 \, \sqrt{2} - 2}\right ) - \frac{21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac{1}{256} \, \log \left (x^{2} + 2 \, x - 1\right ) + \frac{1}{256} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac{5}{64} \, \log \left (x + 1\right ) - \frac{5}{64} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^6 - 7*x^4 + 7*x^2 - 1)^(-2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.291253, size = 324, normalized size = 3.56 \[ -\frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) - 2 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) - 40 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) + 40 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) - 3 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac{\sqrt{2}{\left (x^{2} + 2 \, x + 3\right )} + 4 \, x + 4}{x^{2} + 2 \, x - 1}\right ) - 3 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac{\sqrt{2}{\left (x^{2} - 2 \, x + 3\right )} + 4 \, x - 4}{x^{2} - 2 \, x - 1}\right ) + 4 \, \sqrt{2}{\left (21 \, x^{5} - 140 \, x^{3} + 103 \, x\right )}\right )}}{1024 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^6 - 7*x^4 + 7*x^2 - 1)^(-2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.00064, size = 296, normalized size = 3.25 \[ - \frac{21 x^{5} - 140 x^{3} + 103 x}{128 x^{6} - 896 x^{4} + 896 x^{2} - 128} - \frac{5 \log{\left (x - 1 \right )}}{64} + \frac{5 \log{\left (x + 1 \right )}}{64} + \left (- \frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right ) \log{\left (x - \frac{8071264001}{202624020} - \frac{471550901878784 \left (- \frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{3}}{2979765} + \frac{1299552375287054336 \left (- \frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{5}}{50656005} + \frac{8071264001 \sqrt{2}}{270165360} \right )} + \left (- \frac{3 \sqrt{2}}{1024} - \frac{1}{256}\right ) \log{\left (x - \frac{8071264001 \sqrt{2}}{270165360} - \frac{8071264001}{202624020} + \frac{1299552375287054336 \left (- \frac{3 \sqrt{2}}{1024} - \frac{1}{256}\right )^{5}}{50656005} - \frac{471550901878784 \left (- \frac{3 \sqrt{2}}{1024} - \frac{1}{256}\right )^{3}}{2979765} \right )} + \left (- \frac{3 \sqrt{2}}{1024} + \frac{1}{256}\right ) \log{\left (x - \frac{8071264001 \sqrt{2}}{270165360} + \frac{1299552375287054336 \left (- \frac{3 \sqrt{2}}{1024} + \frac{1}{256}\right )^{5}}{50656005} - \frac{471550901878784 \left (- \frac{3 \sqrt{2}}{1024} + \frac{1}{256}\right )^{3}}{2979765} + \frac{8071264001}{202624020} \right )} + \left (\frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right ) \log{\left (x - \frac{471550901878784 \left (\frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{3}}{2979765} + \frac{1299552375287054336 \left (\frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{5}}{50656005} + \frac{8071264001}{202624020} + \frac{8071264001 \sqrt{2}}{270165360} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**6-7*x**4+7*x**2-1)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.269489, size = 181, normalized size = 1.99 \[ -\frac{3}{1024} \, \sqrt{2}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} + 2 \right |}}\right ) - \frac{3}{1024} \, \sqrt{2}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} - 2 \right |}}\right ) - \frac{21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac{1}{256} \,{\rm ln}\left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) + \frac{1}{256} \,{\rm ln}\left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac{5}{64} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{5}{64} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^6 - 7*x^4 + 7*x^2 - 1)^(-2),x, algorithm="giac")
[Out]