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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]