Integrand size = 23, antiderivative size = 82 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\frac {a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac {\log \left (1+\frac {d}{e x^2}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{4 d^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^2} \]
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Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2385, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=-\frac {\log \left (\frac {d}{e x^2}+1\right ) \left (2 a+2 b \log \left (c x^n\right )-b n\right )}{4 d^2}+\frac {a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^2} \]
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Rule 2379
Rule 2385
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac {\int \frac {-2 a+b n-2 b \log \left (c x^n\right )}{x \left (d+e x^2\right )} \, dx}{2 d} \\ & = \frac {a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac {\log \left (1+\frac {d}{e x^2}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{4 d^2}+\frac {(b n) \int \frac {\log \left (1+\frac {d}{e x^2}\right )}{x} \, dx}{2 d^2} \\ & = \frac {a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac {\log \left (1+\frac {d}{e x^2}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{4 d^2}+\frac {b n \text {Li}_2\left (-\frac {d}{e x^2}\right )}{4 d^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.40 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\frac {a-b n \log (x)+b \log \left (c x^n\right )}{2 d^2+2 d e x^2}+\frac {\log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d^2}-\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )}{2 d^2}+\frac {b n \left (\frac {\sqrt {e} x \log (x)}{i \sqrt {d}-\sqrt {e} x}-\frac {\sqrt {e} x \log (x)}{i \sqrt {d}+\sqrt {e} x}+2 \log ^2(x)+\log \left (i \sqrt {d}-\sqrt {e} x\right )+\log \left (i \sqrt {d}+\sqrt {e} x\right )-2 \left (\log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )-2 \left (\log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )\right )}{4 d^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.59 (sec) , antiderivative size = 338, normalized size of antiderivative = 4.12
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right )}{2 d \left (e \,x^{2}+d \right )}-\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}+\frac {b n \ln \left (e \,x^{2}+d \right )}{4 d^{2}}-\frac {b n \ln \left (x \right )}{2 d^{2}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{2}}+\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e \left (-\frac {d}{e \left (e \,x^{2}+d \right )}+\frac {\ln \left (e \,x^{2}+d \right )}{e}\right )}{2 d^{2}}+\frac {\ln \left (x \right )}{d^{2}}\right )\) | \(338\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (e\,x^2+d\right )}^2} \,d x \]
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