Integrand size = 14, antiderivative size = 73 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=-\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\operatorname {LogIntegral}\left (c \left (a+b x^2\right )\right )}{4 b c} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2504, 2436, 2334, 2335} \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=\frac {\operatorname {LogIntegral}\left (c \left (b x^2+a\right )\right )}{4 b c}-\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )} \]
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Rule 2334
Rule 2335
Rule 2436
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\log ^3(c (a+b x))} \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{\log ^3(c x)} \, dx,x,a+b x^2\right )}{2 b} \\ & = -\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\log ^2(c x)} \, dx,x,a+b x^2\right )}{4 b} \\ & = -\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{4 b} \\ & = -\frac {a+b x^2}{4 b \log ^2\left (c \left (a+b x^2\right )\right )}-\frac {a+b x^2}{4 b \log \left (c \left (a+b x^2\right )\right )}+\frac {\text {li}\left (c \left (a+b x^2\right )\right )}{4 b c} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=\frac {-\frac {\left (a+b x^2\right ) \left (1+\log \left (c \left (a+b x^2\right )\right )\right )}{\log ^2\left (c \left (a+b x^2\right )\right )}+\frac {\operatorname {LogIntegral}\left (c \left (a+b x^2\right )\right )}{c}}{4 b} \]
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Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {c \left (b \,x^{2}+a \right )}{2 {\ln \left (c \left (b \,x^{2}+a \right )\right )}^{2}}-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2}}{2 b c}\) | \(70\) |
default | \(\frac {-\frac {c \left (b \,x^{2}+a \right )}{2 {\ln \left (c \left (b \,x^{2}+a \right )\right )}^{2}}-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2}}{2 b c}\) | \(70\) |
risch | \(-\frac {\ln \left (c \left (b \,x^{2}+a \right )\right ) b \,x^{2}+b \,x^{2}+\ln \left (c \left (b \,x^{2}+a \right )\right ) a +a}{4 b {\ln \left (c \left (b \,x^{2}+a \right )\right )}^{2}}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{4 b c}\) | \(75\) |
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=-\frac {b c x^{2} - \log \left (b c x^{2} + a c\right )^{2} \operatorname {log\_integral}\left (b c x^{2} + a c\right ) + a c + {\left (b c x^{2} + a c\right )} \log \left (b c x^{2} + a c\right )}{4 \, b c \log \left (b c x^{2} + a c\right )^{2}} \]
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Time = 0.73 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=\frac {\begin {cases} \frac {x^{2}}{2 \log {\left (a c \right )}} & \text {for}\: b = 0 \\0 & \text {for}\: c = 0 \\\frac {\operatorname {Ei}{\left (\log {\left (a c + b c x^{2} \right )} \right )}}{2 b c} & \text {otherwise} \end {cases}}{2} + \frac {- a - b x^{2} + \left (- a - b x^{2}\right ) \log {\left (c \left (a + b x^{2}\right ) \right )}}{4 b \log {\left (c \left (a + b x^{2}\right ) \right )}^{2}} \]
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\[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=\int { \frac {x}{\log \left ({\left (b x^{2} + a\right )} c\right )^{3}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=-\frac {\frac {b c x^{2} + a c}{\log \left (b c x^{2} + a c\right )} + \frac {b c x^{2} + a c}{\log \left (b c x^{2} + a c\right )^{2}} - {\rm Ei}\left (\log \left (b c x^{2} + a c\right )\right )}{4 \, b c} \]
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Time = 1.48 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {x}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx=\frac {\mathrm {logint}\left (c\,\left (b\,x^2+a\right )\right )}{4\,b\,c}-\frac {\frac {a\,c}{4}+\ln \left (c\,\left (b\,x^2+a\right )\right )\,\left (\frac {b\,c\,x^2}{4}+\frac {a\,c}{4}\right )+\frac {b\,c\,x^2}{4}}{b\,c\,{\ln \left (c\,\left (b\,x^2+a\right )\right )}^2} \]
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