Optimal. Leaf size=54 \[ \frac{1}{4 a^2 \left (a+c x^4\right )}-\frac{\log \left (a+c x^4\right )}{4 a^3}+\frac{\log (x)}{a^3}+\frac{1}{8 a \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.0366971, antiderivative size = 54, 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} \[ \frac{1}{4 a^2 \left (a+c x^4\right )}-\frac{\log \left (a+c x^4\right )}{4 a^3}+\frac{\log (x)}{a^3}+\frac{1}{8 a \left (a+c x^4\right )^2} \]
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 )^3} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (a+c x)^3} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{c}{a (a+c x)^3}-\frac{c}{a^2 (a+c x)^2}-\frac{c}{a^3 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{8 a \left (a+c x^4\right )^2}+\frac{1}{4 a^2 \left (a+c x^4\right )}+\frac{\log (x)}{a^3}-\frac{\log \left (a+c x^4\right )}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.0295214, size = 43, normalized size = 0.8 \[ \frac{\frac{a \left (3 a+2 c x^4\right )}{\left (a+c x^4\right )^2}-2 \log \left (a+c x^4\right )+8 \log (x)}{8 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 49, normalized size = 0.9 \begin{align*}{\frac{1}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{1}{4\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01359, size = 81, normalized size = 1.5 \begin{align*} \frac{2 \, c x^{4} + 3 \, a}{8 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} - \frac{\log \left (c x^{4} + a\right )}{4 \, a^{3}} + \frac{\log \left (x^{4}\right )}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90598, size = 196, normalized size = 3.63 \begin{align*} \frac{2 \, a c x^{4} + 3 \, a^{2} - 2 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right ) + 8 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (x\right )}{8 \,{\left (a^{3} c^{2} x^{8} + 2 \, a^{4} c x^{4} + a^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.19497, size = 56, normalized size = 1.04 \begin{align*} \frac{3 a + 2 c x^{4}}{8 a^{4} + 16 a^{3} c x^{4} + 8 a^{2} c^{2} x^{8}} + \frac{\log{\left (x \right )}}{a^{3}} - \frac{\log{\left (\frac{a}{c} + x^{4} \right )}}{4 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09514, size = 80, normalized size = 1.48 \begin{align*} \frac{\log \left (x^{4}\right )}{4 \, a^{3}} - \frac{\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{3}} + \frac{3 \, c^{2} x^{8} + 8 \, a c x^{4} + 6 \, a^{2}}{8 \,{\left (c x^{4} + a\right )}^{2} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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