Optimal. Leaf size=38 \[ -\frac{\log \left (a+c x^4\right )}{4 a^2}+\frac{\log (x)}{a^2}+\frac{1}{4 a \left (a+c x^4\right )} \]
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Rubi [A] time = 0.0258131, 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, Rules used = {266, 44} \[ -\frac{\log \left (a+c x^4\right )}{4 a^2}+\frac{\log (x)}{a^2}+\frac{1}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+c x^4\right )^2} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (a+c x)^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{c}{a (a+c x)^2}-\frac{c}{a^2 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{4 a \left (a+c x^4\right )}+\frac{\log (x)}{a^2}-\frac{\log \left (a+c x^4\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.0125719, 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.
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Maple [A] time = 0.013, size = 35, normalized size = 0.9 \begin{align*}{\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00331, size = 50, normalized size = 1.32 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65208, size = 108, normalized size = 2.84 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.798143, size = 34, normalized size = 0.89 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10544, size = 63, normalized size = 1.66 \begin{align*} \frac{\log \left (x^{4}\right )}{4 \, a^{2}} - \frac{\log \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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