3.1325 \(\int \frac{1}{x \left (a+b x^6\right )} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^6\right )}{6 a} \]

[Out]

Log[x]/a - Log[a + b*x^6]/(6*a)

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Rubi [A]  time = 0.0331682, 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 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^6\right )}{6 a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^6)),x]

[Out]

Log[x]/a - Log[a + b*x^6]/(6*a)

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Rubi in Sympy [A]  time = 5.79293, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{6} \right )}}{6 a} - \frac{\log{\left (a + b x^{6} \right )}}{6 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**6+a),x)

[Out]

log(x**6)/(6*a) - log(a + b*x**6)/(6*a)

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Mathematica [A]  time = 0.0108346, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^6\right )}{6 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^6)),x]

[Out]

Log[x]/a - Log[a + b*x^6]/(6*a)

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Maple [A]  time = 0.006, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{6}+a \right ) }{6\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^6+a),x)

[Out]

ln(x)/a-1/6*ln(b*x^6+a)/a

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Maxima [A]  time = 1.44255, size = 31, normalized size = 1.41 \[ -\frac{\log \left (b x^{6} + a\right )}{6 \, a} + \frac{\log \left (x^{6}\right )}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*x),x, algorithm="maxima")

[Out]

-1/6*log(b*x^6 + a)/a + 1/6*log(x^6)/a

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Fricas [A]  time = 0.218683, size = 24, normalized size = 1.09 \[ -\frac{\log \left (b x^{6} + a\right ) - 6 \, \log \left (x\right )}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*x),x, algorithm="fricas")

[Out]

-1/6*(log(b*x^6 + a) - 6*log(x))/a

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Sympy [A]  time = 0.856905, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{6} \right )}}{6 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**6+a),x)

[Out]

log(x)/a - log(a/b + x**6)/(6*a)

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GIAC/XCAS [A]  time = 0.220296, size = 32, normalized size = 1.45 \[ \frac{{\rm ln}\left (x^{6}\right )}{6 \, a} - \frac{{\rm ln}\left ({\left | b x^{6} + a \right |}\right )}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*x),x, algorithm="giac")

[Out]

1/6*ln(x^6)/a - 1/6*ln(abs(b*x^6 + a))/a