Optimal. Leaf size=49 \[ \frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0354551, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 - x^4)^(-1) - Sqrt[a*x^6]/(x*(1 - x^4)),x]
[Out]
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Rubi in Sympy [A] time = 14.4498, size = 42, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{\operatorname{atanh}{\left (x \right )}}{2} + \frac{\sqrt{a x^{6}} \operatorname{atan}{\left (x \right )}}{2 x^{3}} - \frac{\sqrt{a x^{6}} \operatorname{atanh}{\left (x \right )}}{2 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**4+1)-(a*x**6)**(1/2)/x/(-x**4+1),x)
[Out]
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Mathematica [A] time = 0.13873, size = 0, normalized size = 0. \[ \int \left (\frac{1}{1-x^4}-\frac{\sqrt{a x^6}}{x \left (1-x^4\right )}\right ) \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(1 - x^4)^(-1) - Sqrt[a*x^6]/(x*(1 - x^4)),x]
[Out]
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Maple [A] time = 0.005, size = 37, normalized size = 0.8 \[{\frac{{\it Artanh} \left ( x \right ) }{2}}+{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( -1+x \right ) -\ln \left ( 1+x \right ) +2\,\arctan \left ( x \right ) }{4\,{x}^{3}}\sqrt{a{x}^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^4+1)-(a*x^6)^(1/2)/x/(-x^4+1),x)
[Out]
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Maxima [A] time = 0.777124, size = 57, normalized size = 1.16 \[ \frac{1}{2} \, \sqrt{a} \arctan \left (x\right ) - \frac{1}{4} \, \sqrt{a} \log \left (x + 1\right ) + \frac{1}{4} \, \sqrt{a} \log \left (x - 1\right ) + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^4 - 1) + sqrt(a*x^6)/((x^4 - 1)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.292584, size = 1, normalized size = 0.02 \[ \left [\frac{x^{3} \sqrt{-\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}} \log \left (\frac{{\left (a - 1\right )} x^{4} -{\left (a - 1\right )} x^{2} - 2 \,{\left (x^{3} - \sqrt{a x^{6}}\right )} \sqrt{-\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}}}{x^{4} + x^{2}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt{a x^{6}}{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}, \frac{2 \, x^{3} \sqrt{\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}} \arctan \left (-\frac{{\left (a - 1\right )} x^{4}}{{\left (x^{3} - \sqrt{a x^{6}}\right )} \sqrt{\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt{a x^{6}}{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^4 - 1) + sqrt(a*x^6)/((x^4 - 1)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{x^{5} - x}\, dx - \int \left (- \frac{\sqrt{a x^{6}}}{x^{5} - x}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**4+1)-(a*x**6)**(1/2)/x/(-x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.26123, size = 65, normalized size = 1.33 \[ \frac{1}{4} \,{\left (2 \, \arctan \left (x\right ){\rm sign}\left (x\right ) -{\rm ln}\left ({\left | x + 1 \right |}\right ){\rm sign}\left (x\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ){\rm sign}\left (x\right )\right )} \sqrt{a} + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^4 - 1) + sqrt(a*x^6)/((x^4 - 1)*x),x, algorithm="giac")
[Out]