3.899 \(\int \frac{x}{a-b+2 a x^2+a x^4} \, dx\)

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]

-ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]]/(2*Sqrt[a]*Sqrt[b])

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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]

-ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]]/(2*Sqrt[a]*Sqrt[b])

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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]

-atanh(sqrt(a)*(x**2 + 1)/sqrt(b))/(2*sqrt(a)*sqrt(b))

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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]

-ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]]/(2*Sqrt[a]*Sqrt[b])

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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]

-1/2/(a*b)^(1/2)*arctanh(1/2*(2*a*x^2+2*a)/(a*b)^(1/2))

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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]

Exception raised: ValueError

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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]

[1/4*log(-(2*a*b*x^2 + 2*a*b - (a*x^4 + 2*a*x^2 + a + b)*sqrt(a*b))/(a*x^4 + 2*a
*x^2 + a - b))/sqrt(a*b), 1/2*arctan(b/(sqrt(-a*b)*(x^2 + 1)))/sqrt(-a*b)]

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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]

sqrt(1/(a*b))*log(-b*sqrt(1/(a*b)) + x**2 + 1)/4 - sqrt(1/(a*b))*log(b*sqrt(1/(a
*b)) + x**2 + 1)/4

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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]

1/2*arctan((a*x^2 + a)/sqrt(-a*b))/sqrt(-a*b)