3.316 \(\int \frac{1}{x \left (a+b x^4+c x^8\right )} \, dx\)

Optimal. Leaf size=69 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^4+c x^8\right )}{8 a}+\frac{\log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.140544, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^4+c x^8\right )}{8 a}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^4 + c*x^8)),x]

[Out]

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

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Rubi in Sympy [A]  time = 27.7409, size = 63, normalized size = 0.91 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{4}}{\sqrt{- 4 a c + b^{2}}} \right )}}{4 a \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (x^{4} \right )}}{4 a} - \frac{\log{\left (a + b x^{4} + c x^{8} \right )}}{8 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(c*x**8+b*x**4+a),x)

[Out]

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

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Mathematica [C]  time = 0.0376716, size = 66, normalized size = 0.96 \[ \frac{\log (x)}{a}-\frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right ]}{4 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^4 + c*x^8)),x]

[Out]

Log[x]/a - RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[x - #1] + c*Log[x - #1]*#1^4)/
(b + 2*c*#1^4) & ]/(4*a)

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Maple [A]  time = 0.008, size = 66, normalized size = 1. \[ -{\frac{\ln \left ( c{x}^{8}+b{x}^{4}+a \right ) }{8\,a}}-{\frac{b}{4\,a}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{\ln \left ( x \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^8+b*x^4+a),x)

[Out]

-1/8*ln(c*x^8+b*x^4+a)/a-1/4/a*b/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+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^8 + b*x^4 + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.341948, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + b^{3} - 4 \, a b c +{\left (2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (\log \left (c x^{8} + b x^{4} + a\right ) - 8 \, \log \left (x\right )\right )}}{8 \, \sqrt{b^{2} - 4 \, a c} a}, -\frac{2 \, b \arctan \left (-\frac{{\left (2 \, c x^{4} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (\log \left (c x^{8} + b x^{4} + a\right ) - 8 \, \log \left (x\right )\right )}}{8 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x),x, algorithm="fricas")

[Out]

[1/8*(b*log((2*(b^2*c - 4*a*c^2)*x^4 + b^3 - 4*a*b*c + (2*c^2*x^8 + 2*b*c*x^4 +
b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^8 + b*x^4 + a)) - sqrt(b^2 - 4*a*c)*(log(c*
x^8 + b*x^4 + a) - 8*log(x)))/(sqrt(b^2 - 4*a*c)*a), -1/8*(2*b*arctan(-(2*c*x^4
+ b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + sqrt(-b^2 + 4*a*c)*(log(c*x^8 + b*x^4 +
a) - 8*log(x)))/(sqrt(-b^2 + 4*a*c)*a)]

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Sympy [A]  time = 18.4928, size = 253, normalized size = 3.67 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac{1}{8 a}\right ) \log{\left (x^{4} + \frac{- 16 a^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac{1}{8 a}\right ) + 4 a b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac{1}{8 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac{1}{8 a}\right ) \log{\left (x^{4} + \frac{- 16 a^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac{1}{8 a}\right ) + 4 a b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac{1}{8 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \frac{\log{\left (x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(c*x**8+b*x**4+a),x)

[Out]

(-b*sqrt(-4*a*c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a))*log(x**4 + (-16*a**2*c*(
-b*sqrt(-4*a*c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a)) + 4*a*b**2*(-b*sqrt(-4*a*
c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a)) - 2*a*c + b**2)/(b*c)) + (b*sqrt(-4*a*
c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a))*log(x**4 + (-16*a**2*c*(b*sqrt(-4*a*c
+ b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a)) + 4*a*b**2*(b*sqrt(-4*a*c + b**2)/(8*a*(
4*a*c - b**2)) - 1/(8*a)) - 2*a*c + b**2)/(b*c)) + log(x)/a

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GIAC/XCAS [A]  time = 0.307605, size = 92, normalized size = 1.33 \[ -\frac{b \arctan \left (\frac{2 \, c x^{4} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} a} - \frac{{\rm ln}\left (c x^{8} + b x^{4} + a\right )}{8 \, a} + \frac{{\rm ln}\left (x^{4}\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x),x, algorithm="giac")

[Out]

-1/4*b*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a) - 1/8*ln(
c*x^8 + b*x^4 + a)/a + 1/4*ln(x^4)/a