3.1454 \(\int \frac{1}{x \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^8\right )}{8 a} \]

[Out]

Log[x]/a - Log[a + b*x^8]/(8*a)

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Rubi [A]  time = 0.0371897, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^8\right )}{8 a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^8)),x]

[Out]

Log[x]/a - Log[a + b*x^8]/(8*a)

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Rubi in Sympy [A]  time = 5.65375, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{8} \right )}}{8 a} - \frac{\log{\left (a + b x^{8} \right )}}{8 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**8)/(8*a) - log(a + b*x**8)/(8*a)

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Mathematica [A]  time = 0.0110701, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^8\right )}{8 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^8)),x]

[Out]

Log[x]/a - Log[a + b*x^8]/(8*a)

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Maple [A]  time = 0.005, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{8}+a \right ) }{8\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^8+a),x)

[Out]

ln(x)/a-1/8*ln(b*x^8+a)/a

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Maxima [A]  time = 1.43078, size = 31, normalized size = 1.41 \[ -\frac{\log \left (b x^{8} + a\right )}{8 \, a} + \frac{\log \left (x^{8}\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*log(b*x^8 + a)/a + 1/8*log(x^8)/a

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Fricas [A]  time = 0.223066, size = 24, normalized size = 1.09 \[ -\frac{\log \left (b x^{8} + a\right ) - 8 \, \log \left (x\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(log(b*x^8 + a) - 8*log(x))/a

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Sympy [A]  time = 1.0212, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{8} \right )}}{8 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**8+a),x)

[Out]

log(x)/a - log(a/b + x**8)/(8*a)

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GIAC/XCAS [A]  time = 0.228475, size = 32, normalized size = 1.45 \[ \frac{{\rm ln}\left (x^{8}\right )}{8 \, a} - \frac{{\rm ln}\left ({\left | b x^{8} + a \right |}\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*ln(x^8)/a - 1/8*ln(abs(b*x^8 + a))/a