Optimal. Leaf size=175 \[ \frac{i n \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right )}{\sqrt{a} \sqrt{c} e}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt{a} \sqrt{c} e}+\frac{i n \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{c} e}+\frac{2 n \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c} e} \]
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Rubi [A] time = 0.157902, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {205, 2470, 12, 4920, 4854, 2402, 2315} \[ \frac{i n \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right )}{\sqrt{a} \sqrt{c} e}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt{a} \sqrt{c} e}+\frac{i n \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{c} e}+\frac{2 n \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c} e} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2470
Rule 12
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (a+c x^2\right )^n\right )}{a e+c e x^2} \, 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}-(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) \operatorname{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*}
Mathematica [A] time = 0.0430713, size = 131, normalized size = 0.75 \[ \frac{i n \text{PolyLog}\left (2,\frac{\sqrt{c} x+i \sqrt{a}}{\sqrt{c} x-i \sqrt{a}}\right )+\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (\log \left (d \left (a+c x^2\right )^n\right )+2 n \log \left (\frac{2 i}{-\frac{\sqrt{c} x}{\sqrt{a}}+i}\right )+i n \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{\sqrt{a} \sqrt{c} e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+a \right ) ^{n} \right ) }{ce{x}^{2}+ea}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\log{\left (d \left (a + c x^{2}\right )^{n} \right )}}{a + c x^{2}}\, dx}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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