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

Optimal. Leaf size=70 \[ \frac{b \tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{\log \left (a-b x^2+c x^4\right )}{4 a}+\frac{\log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.160301, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{b \tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{\log \left (a-b x^2+c x^4\right )}{4 a}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.9092, size = 63, normalized size = 0.9 \[ - \frac{b \operatorname{atanh}{\left (\frac{- b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (x^{2} \right )}}{2 a} - \frac{\log{\left (a - b x^{2} + c x^{4} \right )}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.138204, size = 117, normalized size = 1.67 \[ \frac{\left (b-\sqrt{b^2-4 a c}\right ) \log \left (-\sqrt{b^2-4 a c}-b+2 c x^2\right )-\left (\sqrt{b^2-4 a c}+b\right ) \log \left (\sqrt{b^2-4 a c}-b+2 c x^2\right )+4 \log (x) \sqrt{b^2-4 a c}}{4 a \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 69, normalized size = 1. \[ -{\frac{\ln \left ( c{x}^{4}-b{x}^{2}+a \right ) }{4\,a}}+{\frac{b}{2\,a}\arctan \left ({(2\,c{x}^{2}-b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{\ln \left ( x \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4-b*x^2+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271966, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (\frac{b^{3} - 4 \, a b c - 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} - b x^{2} + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (\log \left (c x^{4} - b x^{2} + a\right ) - 4 \, \log \left (x\right )\right )}}{4 \, \sqrt{b^{2} - 4 \, a c} a}, \frac{2 \, b \arctan \left (-\frac{{\left (2 \, c x^{2} - b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left (\log \left (c x^{4} - b x^{2} + a\right ) - 4 \, \log \left (x\right )\right )}}{4 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(b*log((b^3 - 4*a*b*c - 2*(b^2*c - 4*a*c^2)*x^2 + (2*c^2*x^4 - 2*b*c*x^2 +
b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 - b*x^2 + a)) - sqrt(b^2 - 4*a*c)*(log(c*
x^4 - b*x^2 + a) - 4*log(x)))/(sqrt(b^2 - 4*a*c)*a), 1/4*(2*b*arctan(-(2*c*x^2 -
b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - sqrt(-b^2 + 4*a*c)*(log(c*x^4 - b*x^2 +
a) - 4*log(x)))/(sqrt(-b^2 + 4*a*c)*a)]

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Sympy [A]  time = 9.01307, size = 253, normalized size = 3.61 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) \log{\left (x^{2} + \frac{8 a^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) - 2 a b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) + 2 a c - b^{2}}{b c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) \log{\left (x^{2} + \frac{8 a^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) - 2 a b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) + 2 a c - b^{2}}{b c} \right )} + \frac{\log{\left (x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.297119, size = 96, normalized size = 1.37 \[ \frac{b \arctan \left (\frac{2 \, c x^{2} - b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a} - \frac{{\rm ln}\left (c x^{4} - b x^{2} + a\right )}{4 \, a} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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