Integrand size = 25, antiderivative size = 201 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=-\frac {e p \log \left (d+e x^2\right )}{2 f (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (f+g x^2\right )}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {e p \log \left (f+g x^2\right )}{2 f (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{2 f^2} \]
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Time = 0.20 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2525, 46, 2463, 2441, 2352, 2442, 36, 31, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (f+g x^2\right )}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{2 f^2}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )}{2 f^2}-\frac {e p \log \left (d+e x^2\right )}{2 f (e f-d g)}+\frac {e p \log \left (f+g x^2\right )}{2 f (e f-d g)} \]
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Rule 31
Rule 36
Rule 46
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x (f+g x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{f^2 x}-\frac {g \log \left (c (d+e x)^p\right )}{f (f+g x)^2}-\frac {g \log \left (c (d+e x)^p\right )}{f^2 (f+g x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^2\right )}{2 f^2}-\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 f^2}-\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^2\right )}{2 f} \\ & = \frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (f+g x^2\right )}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^2\right )}{2 f^2}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{2 f^2}-\frac {(e p) \text {Subst}\left (\int \frac {1}{(d+e x) (f+g x)} \, dx,x,x^2\right )}{2 f} \\ & = \frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (f+g x^2\right )}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}+\frac {p \text {Li}_2\left (1+\frac {e x^2}{d}\right )}{2 f^2}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 f^2}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 f (e f-d g)}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{f+g x} \, dx,x,x^2\right )}{2 f (e f-d g)} \\ & = -\frac {e p \log \left (d+e x^2\right )}{2 f (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (f+g x^2\right )}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {e p \log \left (f+g x^2\right )}{2 f (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}+\frac {p \text {Li}_2\left (1+\frac {e x^2}{d}\right )}{2 f^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\frac {\frac {e f p \log \left (d+e x^2\right )}{-e f+d g}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2}+\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {e f p \log \left (f+g x^2\right )}{e f-d g}-\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )-p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{2 f^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.56 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.45
method | result | size |
parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (x \right )}{f^{2}}+\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 f \left (g \,x^{2}+f \right )}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 f^{2}}-p e \left (-\frac {\ln \left (e \,x^{2}+d \right )}{2 f \left (d g -e f \right )}+\frac {\ln \left (g \,x^{2}+f \right )}{2 f \left (d g -e f \right )}+\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{e}}{f^{2}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )}{2 f^{2} e}\right )\) | \(492\) |
risch | \(\text {Expression too large to display}\) | \(650\) |
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\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.34 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.98 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=-\frac {1}{2} \, e p {\left (\frac {\log \left (e x^{2} + d\right )}{e f^{2} - d f g} - \frac {\log \left (g x^{2} + f\right )}{e f^{2} - d f g} + \frac {2 \, \log \left (\frac {e x^{2}}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x^{2}}{d}\right )}{e f^{2}} - \frac {\log \left (g x^{2} + f\right ) \log \left (-\frac {e g x^{2} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{2} + e f}{e f - d g}\right )}{e f^{2}}\right )} + \frac {1}{2} \, {\left (\frac {1}{f g x^{2} + f^{2}} - \frac {\log \left (g x^{2} + f\right )}{f^{2}} + \frac {\log \left (x^{2}\right )}{f^{2}}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
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\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{x\,{\left (g\,x^2+f\right )}^2} \,d x \]
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