Integrand size = 20, antiderivative size = 229 \[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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Time = 0.16 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2456, 2441, 2440, 2438} \[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \sqrt {-f}}-\frac {\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \sqrt {-f}} \\ & = \frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} \sqrt {g}}+\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} \sqrt {g}} \\ & = \frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} \sqrt {g}} \\ & = \frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\frac {\log \left (c (d+e x)^p\right ) \left (\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )-\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )\right )-p \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.92 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.71
method | result | size |
risch | \(-\frac {\arctan \left (\frac {2 g \left (e x +d \right )-2 d g}{2 e \sqrt {f g}}\right ) p \ln \left (e x +d \right )}{\sqrt {f g}}+\frac {\arctan \left (\frac {2 g \left (e x +d \right )-2 d g}{2 e \sqrt {f g}}\right ) \ln \left (\left (e x +d \right )^{p}\right )}{\sqrt {f g}}+\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 \sqrt {-f g}}-\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 \sqrt {-f g}}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 \sqrt {-f g}}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 \sqrt {-f g}}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}}\) | \(392\) |
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\[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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\[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\int \frac {\log {\left (c \left (d + e x\right )^{p} \right )}}{f + g x^{2}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.35 \[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\frac {e p {\left (\frac {2 \, \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (e x + d\right )}{e} + \frac {\arctan \left (\frac {{\left (e^{2} x + d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f + d^{2} g}, \frac {d e g x + d^{2} g}{e^{2} f + d^{2} g}\right ) \log \left (g x^{2} + f\right ) - \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {e^{2} g x^{2} + 2 \, d e g x + d^{2} g}{e^{2} f + d^{2} g}\right ) - i \, {\rm Li}_2\left (\frac {d e g x + e^{2} f - {\left (i \, e^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f + 2 i \, d e \sqrt {f} \sqrt {g} - d^{2} g}\right ) + i \, {\rm Li}_2\left (\frac {d e g x + e^{2} f + {\left (i \, e^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f - 2 i \, d e \sqrt {f} \sqrt {g} - d^{2} g}\right )}{e}\right )}}{2 \, \sqrt {f g}} - \frac {p \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (e x + d\right )}{\sqrt {f g}} + \frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left ({\left (e x + d\right )}^{p} c\right )}{\sqrt {f g}} \]
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\[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x\right )}^p\right )}{g\,x^2+f} \,d x \]
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