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 = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 44
Rule 266
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.68, size = 47, normalized size = 1.24 \[ -\frac {{\left (c x^{4} + a\right )} \log \left (c x^{4} + a\right ) - 4 \, {\left (c x^{4} + a\right )} \log \relax (x) - a}{4 \, {\left (a^{2} c x^{4} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 47, normalized size = 1.24 \[ \frac {\log \left (x^{4}\right )}{4 \, a^{2}} - \frac {\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{2}} + \frac {c x^{4} + 2 \, a}{4 \, {\left (c x^{4} + a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 35, normalized size = 0.92 \[ \frac {1}{4 \left (c \,x^{4}+a \right ) a}+\frac {\ln \relax (x )}{a^{2}}-\frac {\ln \left (c \,x^{4}+a \right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 37, normalized size = 0.97 \[ \frac {1}{4 \, {\left (a c x^{4} + a^{2}\right )}} - \frac {\log \left (c x^{4} + a\right )}{4 \, a^{2}} + \frac {\log \left (x^{4}\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 34, normalized size = 0.89 \[ \frac {\ln \relax (x)}{a^2}+\frac {1}{4\,a\,\left (c\,x^4+a\right )}-\frac {\ln \left (c\,x^4+a\right )}{4\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 34, normalized size = 0.89 \[ \frac {1}{4 a^{2} + 4 a c x^{4}} + \frac {\log {\relax (x )}}{a^{2}} - \frac {\log {\left (\frac {a}{c} + x^{4} \right )}}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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