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

Optimal. Leaf size=38 \[ -\frac{\log \left (a+c x^4\right )}{4 a^2}+\frac{\log (x)}{a^2}+\frac{1}{4 a \left (a+c x^4\right )} \]

[Out]

1/(4*a*(a + c*x^4)) + Log[x]/a^2 - Log[a + c*x^4]/(4*a^2)

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Rubi [A]  time = 0.0586852, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\log \left (a+c x^4\right )}{4 a^2}+\frac{\log (x)}{a^2}+\frac{1}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + c*x^4)^2),x]

[Out]

1/(4*a*(a + c*x^4)) + Log[x]/a^2 - Log[a + c*x^4]/(4*a^2)

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Rubi in Sympy [A]  time = 8.2986, size = 34, normalized size = 0.89 \[ \frac{1}{4 a \left (a + c x^{4}\right )} + \frac{\log{\left (x^{4} \right )}}{4 a^{2}} - \frac{\log{\left (a + c x^{4} \right )}}{4 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0236803, size = 33, normalized size = 0.87 \[ \frac{\frac{a}{a+c x^4}-\log \left (a+c x^4\right )+4 \log (x)}{4 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 35, normalized size = 0.9 \[{\frac{1}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.42478, size = 50, normalized size = 1.32 \[ \frac{1}{4 \,{\left (a c x^{4} + a^{2}\right )}} - \frac{\log \left (c x^{4} + a\right )}{4 \, a^{2}} + \frac{\log \left (x^{4}\right )}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/(a*c*x^4 + a^2) - 1/4*log(c*x^4 + a)/a^2 + 1/4*log(x^4)/a^2

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Fricas [A]  time = 0.233987, size = 63, normalized size = 1.66 \[ -\frac{{\left (c x^{4} + a\right )} \log \left (c x^{4} + a\right ) - 4 \,{\left (c x^{4} + a\right )} \log \left (x\right ) - a}{4 \,{\left (a^{2} c x^{4} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*((c*x^4 + a)*log(c*x^4 + a) - 4*(c*x^4 + a)*log(x) - a)/(a^2*c*x^4 + a^3)

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Sympy [A]  time = 2.35357, size = 34, normalized size = 0.89 \[ \frac{1}{4 a^{2} + 4 a c x^{4}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (\frac{a}{c} + x^{4} \right )}}{4 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(c*x**4+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223932, size = 63, normalized size = 1.66 \[ \frac{{\rm ln}\left (x^{4}\right )}{4 \, a^{2}} - \frac{{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{2}} + \frac{c x^{4} + 2 \, a}{4 \,{\left (c x^{4} + a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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