3.900 \(\int \frac{1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx\)

Optimal. Leaf size=77 \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a-b)}-\frac{\log \left (a x^4+2 a x^2+a-b\right )}{4 (a-b)}+\frac{\log (x)}{a-b} \]

[Out]

(Sqrt[a]*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*(a - b)*Sqrt[b]) + Log[x]/(a -
b) - Log[a - b + 2*a*x^2 + a*x^4]/(4*(a - b))

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Rubi [A]  time = 0.15787, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a-b)}-\frac{\log \left (a x^4+2 a x^2+a-b\right )}{4 (a-b)}+\frac{\log (x)}{a-b} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a - b + 2*a*x^2 + a*x^4)),x]

[Out]

(Sqrt[a]*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*(a - b)*Sqrt[b]) + Log[x]/(a -
b) - Log[a - b + 2*a*x^2 + a*x^4]/(4*(a - b))

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Rubi in Sympy [A]  time = 31.189, size = 66, normalized size = 0.86 \[ \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 \sqrt{b} \left (a - b\right )} + \frac{\log{\left (x^{2} \right )}}{2 \left (a - b\right )} - \frac{\log{\left (a x^{4} + 2 a x^{2} + a - b \right )}}{4 \left (a - b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a*x**4+2*a*x**2+a-b),x)

[Out]

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

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Mathematica [A]  time = 0.0844579, size = 90, normalized size = 1.17 \[ \frac{\left (\sqrt{a}+\sqrt{b}\right ) \log \left (\sqrt{a} \left (x^2+1\right )-\sqrt{b}\right )+\left (\sqrt{b}-\sqrt{a}\right ) \log \left (\sqrt{a} \left (x^2+1\right )+\sqrt{b}\right )-4 \sqrt{b} \log (x)}{4 \sqrt{b} (b-a)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a - b + 2*a*x^2 + a*x^4)),x]

[Out]

(-4*Sqrt[b]*Log[x] + (Sqrt[a] + Sqrt[b])*Log[-Sqrt[b] + Sqrt[a]*(1 + x^2)] + (-S
qrt[a] + Sqrt[b])*Log[Sqrt[b] + Sqrt[a]*(1 + x^2)])/(4*Sqrt[b]*(-a + b))

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Maple [A]  time = 0.009, size = 71, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{a-b}}-{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a-b \right ) }{4\,a-4\,b}}+{\frac{a}{2\,a-2\,b}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a*x^4+2*a*x^2+a-b),x)

[Out]

ln(x)/(a-b)-1/4*ln(a*x^4+2*a*x^2+a-b)/(a-b)+1/2*a/(a-b)/(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(1/((a*x^4 + 2*a*x^2 + a - b)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284392, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{\frac{a}{b}} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \,{\left (b x^{2} + b\right )} \sqrt{\frac{a}{b}} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \,{\left (a - b\right )}}, -\frac{2 \, \sqrt{-\frac{a}{b}} \arctan \left (\frac{b \sqrt{-\frac{a}{b}}}{a x^{2} + a}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \,{\left (a - b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a - b)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.44558, size = 184, normalized size = 2.39 \[ \left (- \frac{1}{4 \left (a - b\right )} - \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) \log{\left (x^{2} + \frac{4 a b \left (- \frac{1}{4 \left (a - b\right )} - \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a - b\right )} - \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \left (- \frac{1}{4 \left (a - b\right )} + \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) \log{\left (x^{2} + \frac{4 a b \left (- \frac{1}{4 \left (a - b\right )} + \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a - b\right )} + \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \frac{\log{\left (x \right )}}{a - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a*x**4+2*a*x**2+a-b),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.535827, size = 96, normalized size = 1.25 \[ -\frac{a \arctan \left (\frac{a x^{2} + a}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b}{\left (a - b\right )}} - \frac{{\rm ln}\left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \,{\left (a - b\right )}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \,{\left (a - b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a - b)*x),x, algorithm="giac")

[Out]

-1/2*a*arctan((a*x^2 + a)/sqrt(-a*b))/(sqrt(-a*b)*(a - b)) - 1/4*ln(a*x^4 + 2*a*
x^2 + a - b)/(a - b) + 1/2*ln(x^2)/(a - b)