Integrand size = 16, antiderivative size = 35 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=\frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2436, 2332} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=\frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b} \]
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Rule 2332
Rule 2436
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = -\frac {\text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+\frac {b}{x^2}\right )}{2 b} \\ & = \frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=\frac {1}{2} \left (\frac {p}{x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{b}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \left (a +\frac {b}{x^{2}}\right )-\left (a +\frac {b}{x^{2}}\right ) p}{2 b}\) | \(37\) |
default | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \left (a +\frac {b}{x^{2}}\right )-\left (a +\frac {b}{x^{2}}\right ) p}{2 b}\) | \(37\) |
parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{2 x^{2}}-p b \left (-\frac {1}{2 b \,x^{2}}-\frac {a \ln \left (x \right )}{b^{2}}+\frac {a \ln \left (x^{2} a +b \right )}{2 b^{2}}\right )\) | \(54\) |
parallelrisch | \(-\frac {x^{2} \ln \left (c \left (\frac {x^{2} a +b}{x^{2}}\right )^{p}\right ) a^{2} p +\ln \left (c \left (\frac {x^{2} a +b}{x^{2}}\right )^{p}\right ) a b p -a b \,p^{2}}{2 x^{2} a p b}\) | \(67\) |
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=\frac {b p - b \log \left (c\right ) - {\left (a p x^{2} + b p\right )} \log \left (\frac {a x^{2} + b}{x^{2}}\right )}{2 \, b x^{2}} \]
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Time = 0.80 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=\begin {cases} - \frac {a \log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{2 b} + \frac {p}{2 x^{2}} - \frac {\log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{2 x^{2}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=-\frac {1}{2} \, b p {\left (\frac {a \log \left (a x^{2} + b\right )}{b^{2}} - \frac {a \log \left (x^{2}\right )}{b^{2}} - \frac {1}{b x^{2}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{2 \, x^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=-\frac {p {\left (\frac {{\left (a x^{2} + b\right )} \log \left (\frac {a x^{2} + b}{x^{2}}\right )}{x^{2}} - \frac {a x^{2} + b}{x^{2}}\right )} + \frac {{\left (a x^{2} + b\right )} \log \left (c\right )}{x^{2}}}{2 \, b} \]
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Time = 1.39 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx=\frac {p}{2\,x^2}-\frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{2\,x^2}-\frac {a\,p\,\ln \left (a\,x^2+b\right )}{2\,b}+\frac {a\,p\,\ln \left (x\right )}{b} \]
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