Integrand size = 25, antiderivative size = 175 \[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\frac {i n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {2 n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e}+\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}+\frac {i n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e} \]
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Time = 0.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {211, 2520, 12, 5040, 4964, 2449, 2352} \[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}+\frac {i n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {2 n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e}+\frac {i n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{i \sqrt {c} x+\sqrt {a}}\right )}{\sqrt {a} \sqrt {c} e} \]
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Rule 12
Rule 211
Rule 2352
Rule 2449
Rule 2520
Rule 4964
Rule 5040
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}-(2 c n) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c} e \left (a+c x^2\right )} \, dx \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}-\frac {\left (2 \sqrt {c} n\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a+c x^2} \, dx}{\sqrt {a} e} \\ & = \frac {i n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}+\frac {(2 n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {c} x}{\sqrt {a}}} \, dx}{a e} \\ & = \frac {i n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {2 n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}-\frac {(2 n) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {c} x}{\sqrt {a}}}\right )}{1+\frac {c x^2}{a}} \, dx}{a e} \\ & = \frac {i n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {2 n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}+\frac {(2 i n) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {c} x}{\sqrt {a}}}\right )}{\sqrt {a} \sqrt {c} e} \\ & = \frac {i n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {2 n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}+\frac {i n \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.75 \[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (i n \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )+2 n \log \left (\frac {2 i}{i-\frac {\sqrt {c} x}{\sqrt {a}}}\right )+\log \left (d \left (a+c x^2\right )^n\right )\right )+i n \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {c} x}{-i \sqrt {a}+\sqrt {c} x}\right )}{\sqrt {a} \sqrt {c} e} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.60 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.67
method | result | size |
risch | \(-\frac {\arctan \left (\frac {x c}{\sqrt {c a}}\right ) n \ln \left (c \,x^{2}+a \right )}{e \sqrt {c a}}+\frac {\arctan \left (\frac {x c}{\sqrt {c a}}\right ) \ln \left (\left (c \,x^{2}+a \right )^{n}\right )}{e \sqrt {c a}}+\frac {n \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (c \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (c \,x^{2}+a \right )-c \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha c}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{4 e c}+\frac {\left (i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+a \right )^{n}\right ) {\operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (c \,x^{2}+a \right )^{n}\right ) \operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right ) \operatorname {csgn}\left (i d \right )-i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right )}^{3}+i \pi {\operatorname {csgn}\left (i d \left (c \,x^{2}+a \right )^{n}\right )}^{2} \operatorname {csgn}\left (i d \right )+2 \ln \left (d \right )\right ) \arctan \left (\frac {x c}{\sqrt {c a}}\right )}{2 e \sqrt {c a}}\) | \(292\) |
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\[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e} \,d x } \]
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\[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\frac {\int \frac {\log {\left (d \left (a + c x^{2}\right )^{n} \right )}}{a + c x^{2}}\, dx}{e} \]
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\[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e} \,d x } \]
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\[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e} \,d x } \]
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Timed out. \[ \int \frac {\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, dx=\int \frac {\ln \left (d\,{\left (c\,x^2+a\right )}^n\right )}{c\,e\,x^2+a\,e} \,d x \]
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