Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+c x^4\right )}{4 a} \]
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Rubi [A] time = 0.0106017, 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, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{\log \left (a+c x^4\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+c x^4\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (a+c x)} \, dx,x,x^4\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^4\right )}{4 a}-\frac{c \operatorname{Subst}\left (\int \frac{1}{a+c x} \, dx,x,x^4\right )}{4 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+c x^4\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0056128, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+c x^4\right )}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01224, size = 31, normalized size = 1.41 \begin{align*} -\frac{\log \left (c x^{4} + a\right )}{4 \, a} + \frac{\log \left (x^{4}\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69201, size = 49, normalized size = 2.23 \begin{align*} -\frac{\log \left (c x^{4} + a\right ) - 4 \, \log \left (x\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.236794, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{c} + x^{4} \right )}}{4 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12883, size = 32, normalized size = 1.45 \begin{align*} \frac{\log \left (x^{4}\right )}{4 \, a} - \frac{\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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