Optimal. Leaf size=54 \[ -\frac {\log \left (a+c x^4\right )}{4 a^3}+\frac {\log (x)}{a^3}+\frac {1}{4 a^2 \left (a+c x^4\right )}+\frac {1}{8 a \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 54, 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} \[ \frac {1}{4 a^2 \left (a+c x^4\right )}-\frac {\log \left (a+c x^4\right )}{4 a^3}+\frac {\log (x)}{a^3}+\frac {1}{8 a \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
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*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.80 \[ \frac {\frac {a \left (3 a+2 c x^4\right )}{\left (a+c x^4\right )^2}-2 \log \left (a+c x^4\right )+8 \log (x)}{8 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 90, normalized size = 1.67 \[ \frac {2 \, a c x^{4} + 3 \, a^{2} - 2 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right ) + 8 \, {\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \relax (x)}{8 \, {\left (a^{3} c^{2} x^{8} + 2 \, a^{4} c x^{4} + a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 59, normalized size = 1.09 \[ \frac {\log \left (x^{4}\right )}{4 \, a^{3}} - \frac {\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, a^{3}} + \frac {3 \, c^{2} x^{8} + 8 \, a c x^{4} + 6 \, a^{2}}{8 \, {\left (c x^{4} + a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 49, normalized size = 0.91 \[ \frac {1}{8 \left (c \,x^{4}+a \right )^{2} a}+\frac {1}{4 \left (c \,x^{4}+a \right ) a^{2}}+\frac {\ln \relax (x )}{a^{3}}-\frac {\ln \left (c \,x^{4}+a \right )}{4 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 60, normalized size = 1.11 \[ \frac {2 \, c x^{4} + 3 \, a}{8 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} - \frac {\log \left (c x^{4} + a\right )}{4 \, a^{3}} + \frac {\log \left (x^{4}\right )}{4 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 56, normalized size = 1.04 \[ \frac {\ln \relax (x)}{a^3}+\frac {\frac {3}{8\,a}+\frac {c\,x^4}{4\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {\ln \left (c\,x^4+a\right )}{4\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 56, normalized size = 1.04 \[ \frac {3 a + 2 c x^{4}}{8 a^{4} + 16 a^{3} c x^{4} + 8 a^{2} c^{2} x^{8}} + \frac {\log {\relax (x )}}{a^{3}} - \frac {\log {\left (\frac {a}{c} + x^{4} \right )}}{4 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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