Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]
[Out]
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Rubi [A] time = 0.0632392, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[x/(a - b + 2*a*x^2 + a*x^4),x]
[Out]
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Rubi in Sympy [A] time = 9.82348, size = 29, normalized size = 0.94 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 \sqrt{a} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a*x**4+2*a*x**2+a-b),x)
[Out]
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Mathematica [A] time = 0.0148255, size = 31, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a - b + 2*a*x^2 + a*x^4),x]
[Out]
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Maple [A] time = 0.002, size = 26, normalized size = 0.8 \[ -{\frac{1}{2}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a*x^4+2*a*x^2+a-b),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(x/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278685, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \, a b x^{2} + 2 \, a b -{\left (a x^{4} + 2 \, a x^{2} + a + b\right )} \sqrt{a b}}{a x^{4} + 2 \, a x^{2} + a - b}\right )}{4 \, \sqrt{a b}}, \frac{\arctan \left (\frac{b}{\sqrt{-a b}{\left (x^{2} + 1\right )}}\right )}{2 \, \sqrt{-a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.788382, size = 53, normalized size = 1.71 \[ \frac{\sqrt{\frac{1}{a b}} \log{\left (- b \sqrt{\frac{1}{a b}} + x^{2} + 1 \right )}}{4} - \frac{\sqrt{\frac{1}{a b}} \log{\left (b \sqrt{\frac{1}{a b}} + x^{2} + 1 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a*x**4+2*a*x**2+a-b),x)
[Out]
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GIAC/XCAS [A] time = 0.54731, size = 31, normalized size = 1. \[ \frac{\arctan \left (\frac{a x^{2} + a}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="giac")
[Out]