3.901 \(\int \frac{1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx\)

Optimal. Leaf size=97 \[ -\frac{1}{2 x^2 (a-b)}-\frac{\sqrt{a} (a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a-b)^2}+\frac{a \log \left (a x^4+2 a x^2+a-b\right )}{2 (a-b)^2}-\frac{2 a \log (x)}{(a-b)^2} \]

[Out]

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

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Rubi [A]  time = 0.307805, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{1}{2 x^2 (a-b)}-\frac{\sqrt{a} (a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a-b)^2}+\frac{a \log \left (a x^4+2 a x^2+a-b\right )}{2 (a-b)^2}-\frac{2 a \log (x)}{(a-b)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a - b + 2*a*x^2 + a*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 47.232, size = 83, normalized size = 0.86 \[ - \frac{\sqrt{a} \left (a + b\right ) \operatorname{atanh}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 \sqrt{b} \left (a - b\right )^{2}} - \frac{a \log{\left (x^{2} \right )}}{\left (a - b\right )^{2}} + \frac{a \log{\left (a x^{4} + 2 a x^{2} + a - b \right )}}{2 \left (a - b\right )^{2}} - \frac{1}{2 x^{2} \left (a - b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a*x**4+2*a*x**2+a-b),x)

[Out]

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

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Mathematica [A]  time = 0.21427, size = 146, normalized size = 1.51 \[ \frac{-8 a \sqrt{b} x^2 \log (x)+\sqrt{a} x^2 \left (\sqrt{a}+\sqrt{b}\right )^2 \log \left (\sqrt{a} \left (x^2+1\right )-\sqrt{b}\right )-\left (\sqrt{a}-\sqrt{b}\right ) \left (\left (a x^2-\sqrt{a} \sqrt{b} x^2\right ) \log \left (\sqrt{a} \left (x^2+1\right )+\sqrt{b}\right )+2 \left (\sqrt{a} \sqrt{b}+b\right )\right )}{4 \sqrt{b} x^2 (a-b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a - b + 2*a*x^2 + a*x^4)),x]

[Out]

(-8*a*Sqrt[b]*x^2*Log[x] + Sqrt[a]*(Sqrt[a] + Sqrt[b])^2*x^2*Log[-Sqrt[b] + Sqrt
[a]*(1 + x^2)] - (Sqrt[a] - Sqrt[b])*(2*(Sqrt[a]*Sqrt[b] + b) + (a*x^2 - Sqrt[a]
*Sqrt[b]*x^2)*Log[Sqrt[b] + Sqrt[a]*(1 + x^2)]))/(4*(a - b)^2*Sqrt[b]*x^2)

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Maple [A]  time = 0.014, size = 122, normalized size = 1.3 \[ -{\frac{1}{ \left ( 2\,a-2\,b \right ){x}^{2}}}-2\,{\frac{a\ln \left ( x \right ) }{ \left ( a-b \right ) ^{2}}}+{\frac{a\ln \left ( a{x}^{4}+2\,a{x}^{2}+a-b \right ) }{2\, \left ( a-b \right ) ^{2}}}-{\frac{{a}^{2}}{2\, \left ( a-b \right ) ^{2}}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{ab}{2\, \left ( a-b \right ) ^{2}}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a*x^4+2*a*x^2+a-b),x)

[Out]

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

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290063, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a + b\right )} x^{2} \sqrt{\frac{a}{b}} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \,{\left (b x^{2} + b\right )} \sqrt{\frac{a}{b}} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + 2 \, a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 8 \, a x^{2} \log \left (x\right ) - 2 \, a + 2 \, b}{4 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} x^{2}}, \frac{{\left (a + b\right )} x^{2} \sqrt{-\frac{a}{b}} \arctan \left (\frac{b \sqrt{-\frac{a}{b}}}{a x^{2} + a}\right ) + a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, a x^{2} \log \left (x\right ) - a + b}{2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a - b)*x^3),x, algorithm="fricas")

[Out]

[1/4*((a + b)*x^2*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)) + 2*a*x^2*log(a*x^4 + 2*a*x^2 + a - b) - 8*a*x^2*
log(x) - 2*a + 2*b)/((a^2 - 2*a*b + b^2)*x^2), 1/2*((a + b)*x^2*sqrt(-a/b)*arcta
n(b*sqrt(-a/b)/(a*x^2 + a)) + a*x^2*log(a*x^4 + 2*a*x^2 + a - b) - 4*a*x^2*log(x
) - a + b)/((a^2 - 2*a*b + b^2)*x^2)]

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Sympy [A]  time = 17.3448, size = 372, normalized size = 3.84 \[ - \frac{2 a \log{\left (x \right )}}{\left (a - b\right )^{2}} + \left (\frac{a}{2 \left (a - b\right )^{2}} - \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- 4 a^{2} b \left (\frac{a}{2 \left (a - b\right )^{2}} - \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac{a}{2 \left (a - b\right )^{2}} - \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + 3 a b - 4 b^{3} \left (\frac{a}{2 \left (a - b\right )^{2}} - \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right )}{a^{2} + a b} \right )} + \left (\frac{a}{2 \left (a - b\right )^{2}} + \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- 4 a^{2} b \left (\frac{a}{2 \left (a - b\right )^{2}} + \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac{a}{2 \left (a - b\right )^{2}} + \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + 3 a b - 4 b^{3} \left (\frac{a}{2 \left (a - b\right )^{2}} + \frac{\sqrt{a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right )}{a^{2} + a b} \right )} - \frac{1}{x^{2} \left (2 a - 2 b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a*x**4+2*a*x**2+a-b),x)

[Out]

-2*a*log(x)/(a - b)**2 + (a/(2*(a - b)**2) - sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*
b + b**2)))*log(x**2 + (-4*a**2*b*(a/(2*(a - b)**2) - sqrt(a*b)*(a + b)/(4*b*(a*
*2 - 2*a*b + b**2))) + a**2 + 8*a*b**2*(a/(2*(a - b)**2) - sqrt(a*b)*(a + b)/(4*
b*(a**2 - 2*a*b + b**2))) + 3*a*b - 4*b**3*(a/(2*(a - b)**2) - sqrt(a*b)*(a + b)
/(4*b*(a**2 - 2*a*b + b**2))))/(a**2 + a*b)) + (a/(2*(a - b)**2) + sqrt(a*b)*(a
+ b)/(4*b*(a**2 - 2*a*b + b**2)))*log(x**2 + (-4*a**2*b*(a/(2*(a - b)**2) + sqrt
(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2))) + a**2 + 8*a*b**2*(a/(2*(a - b)**2) +
sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2))) + 3*a*b - 4*b**3*(a/(2*(a - b)**
2) + sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2))))/(a**2 + a*b)) - 1/(x**2*(2*
a - 2*b))

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GIAC/XCAS [A]  time = 0.551049, size = 170, normalized size = 1.75 \[ \frac{a{\rm ln}\left (a x^{4} + 2 \, a x^{2} + a - b\right )}{2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac{a{\rm ln}\left (x^{2}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{{\left (a^{2} + a b\right )} \arctan \left (\frac{a x^{2} + a}{\sqrt{-a b}}\right )}{2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt{-a b}} + \frac{2 \, a x^{2} - a + b}{2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a - b)*x^3),x, algorithm="giac")

[Out]

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