Optimal. Leaf size=47 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt{a+1}}\right )}{2 \sqrt{a+1}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1-a}}\right )}{2 \sqrt{1-a}} \]
[Out]
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Rubi [A] time = 0.0516353, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt{a+1}}\right )}{2 \sqrt{a+1}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1-a}}\right )}{2 \sqrt{1-a}} \]
Antiderivative was successfully verified.
[In] Int[(-1 + a^2 + 2*a*x^2 + x^4)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 8.92252, size = 37, normalized size = 0.79 \[ - \frac{\operatorname{atan}{\left (\frac{x}{\sqrt{a + 1}} \right )}}{2 \sqrt{a + 1}} - \frac{\operatorname{atanh}{\left (\frac{x}{\sqrt{- a + 1}} \right )}}{2 \sqrt{- a + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**4+2*a*x**2+a**2-1),x)
[Out]
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Mathematica [A] time = 0.0388376, size = 43, normalized size = 0.91 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{a-1}}\right )}{2 \sqrt{a-1}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{a+1}}\right )}{2 \sqrt{a+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + a^2 + 2*a*x^2 + x^4)^(-1),x]
[Out]
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Maple [A] time = 0.015, size = 32, normalized size = 0.7 \[ -{\frac{1}{2}\arctan \left ({x{\frac{1}{\sqrt{1+a}}}} \right ){\frac{1}{\sqrt{1+a}}}}+{\frac{1}{2}\arctan \left ({x{\frac{1}{\sqrt{a-1}}}} \right ){\frac{1}{\sqrt{a-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^4+2*a*x^2+a^2-1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 + 2*a*x^2 + a^2 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278277, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-a + 1} \log \left (-\frac{2 \,{\left (a + 1\right )} x -{\left (x^{2} - a - 1\right )} \sqrt{-a - 1}}{x^{2} + a + 1}\right ) + \sqrt{-a - 1} \log \left (\frac{2 \,{\left (a - 1\right )} x +{\left (x^{2} - a + 1\right )} \sqrt{-a + 1}}{x^{2} + a - 1}\right )}{4 \, \sqrt{-a + 1} \sqrt{-a - 1}}, \frac{2 \, \sqrt{-a - 1} \arctan \left (\frac{x}{\sqrt{a - 1}}\right ) + \sqrt{a - 1} \log \left (-\frac{2 \,{\left (a + 1\right )} x -{\left (x^{2} - a - 1\right )} \sqrt{-a - 1}}{x^{2} + a + 1}\right )}{4 \, \sqrt{a - 1} \sqrt{-a - 1}}, -\frac{2 \, \sqrt{-a + 1} \arctan \left (\frac{x}{\sqrt{a + 1}}\right ) - \sqrt{a + 1} \log \left (\frac{2 \,{\left (a - 1\right )} x +{\left (x^{2} - a + 1\right )} \sqrt{-a + 1}}{x^{2} + a - 1}\right )}{4 \, \sqrt{a + 1} \sqrt{-a + 1}}, -\frac{\sqrt{a - 1} \arctan \left (\frac{x}{\sqrt{a + 1}}\right ) - \sqrt{a + 1} \arctan \left (\frac{x}{\sqrt{a - 1}}\right )}{2 \, \sqrt{a + 1} \sqrt{a - 1}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 + 2*a*x^2 + a^2 - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.46282, size = 257, normalized size = 5.47 \[ \frac{\sqrt{- \frac{1}{a - 1}} \log{\left (- a^{3} \left (- \frac{1}{a - 1}\right )^{\frac{3}{2}} - a^{2} \sqrt{- \frac{1}{a - 1}} + a \left (- \frac{1}{a - 1}\right )^{\frac{3}{2}} + x - \sqrt{- \frac{1}{a - 1}} \right )}}{4} - \frac{\sqrt{- \frac{1}{a - 1}} \log{\left (a^{3} \left (- \frac{1}{a - 1}\right )^{\frac{3}{2}} + a^{2} \sqrt{- \frac{1}{a - 1}} - a \left (- \frac{1}{a - 1}\right )^{\frac{3}{2}} + x + \sqrt{- \frac{1}{a - 1}} \right )}}{4} + \frac{\sqrt{- \frac{1}{a + 1}} \log{\left (- a^{3} \left (- \frac{1}{a + 1}\right )^{\frac{3}{2}} - a^{2} \sqrt{- \frac{1}{a + 1}} + a \left (- \frac{1}{a + 1}\right )^{\frac{3}{2}} + x - \sqrt{- \frac{1}{a + 1}} \right )}}{4} - \frac{\sqrt{- \frac{1}{a + 1}} \log{\left (a^{3} \left (- \frac{1}{a + 1}\right )^{\frac{3}{2}} + a^{2} \sqrt{- \frac{1}{a + 1}} - a \left (- \frac{1}{a + 1}\right )^{\frac{3}{2}} + x + \sqrt{- \frac{1}{a + 1}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**4+2*a*x**2+a**2-1),x)
[Out]
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GIAC/XCAS [A] time = 0.286117, size = 42, normalized size = 0.89 \[ -\frac{\arctan \left (\frac{x}{\sqrt{a + 1}}\right )}{2 \, \sqrt{a + 1}} + \frac{\arctan \left (\frac{x}{\sqrt{a - 1}}\right )}{2 \, \sqrt{a - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^4 + 2*a*x^2 + a^2 - 1),x, algorithm="giac")
[Out]