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

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+c x^4\right )}{4 a} \]

[Out]

Log[x]/a - Log[a + c*x^4]/(4*a)

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Rubi [A]  time = 0.0331179, 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 \[ \frac{\log (x)}{a}-\frac{\log \left (a+c x^4\right )}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + c*x^4)),x]

[Out]

Log[x]/a - Log[a + c*x^4]/(4*a)

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Rubi in Sympy [A]  time = 5.44952, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{4} \right )}}{4 a} - \frac{\log{\left (a + c x^{4} \right )}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**4)/(4*a) - log(a + c*x**4)/(4*a)

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Mathematica [A]  time = 0.0107981, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+c x^4\right )}{4 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + c*x^4)),x]

[Out]

Log[x]/a - Log[a + c*x^4]/(4*a)

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Maple [A]  time = 0.007, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+a),x)

[Out]

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

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Maxima [A]  time = 1.43191, size = 31, normalized size = 1.41 \[ -\frac{\log \left (c x^{4} + a\right )}{4 \, a} + \frac{\log \left (x^{4}\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*log(c*x^4 + a)/a + 1/4*log(x^4)/a

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Fricas [A]  time = 0.226576, size = 24, normalized size = 1.09 \[ -\frac{\log \left (c x^{4} + a\right ) - 4 \, \log \left (x\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(log(c*x^4 + a) - 4*log(x))/a

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Sympy [A]  time = 0.665316, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{c} + x^{4} \right )}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(c*x**4+a),x)

[Out]

log(x)/a - log(a/c + x**4)/(4*a)

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GIAC/XCAS [A]  time = 0.222681, size = 32, normalized size = 1.45 \[ \frac{{\rm ln}\left (x^{4}\right )}{4 \, a} - \frac{{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*ln(x^4)/a - 1/4*ln(abs(c*x^4 + a))/a