Optimal. Leaf size=509 \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}} \]
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Rubi [A] time = 0.611818, antiderivative size = 509, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2330, 2318, 2317, 2391, 2374, 6589} \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 2330
Rule 2318
Rule 2317
Rule 2391
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-d e-e^2 x^2} \, dx}{2 d}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{e \int \left (-\frac{\sqrt{-d} \left (a+b \log \left (c x^n\right )\right )^2}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \log \left (c x^n\right )\right )^2}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d}-\frac{\left (b \sqrt{e} n\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{-d} \sqrt{e}-e x} \, dx}{2 (-d)^{3/2}}-\frac{\left (b \sqrt{e} n\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{-d} \sqrt{e}+e x} \, dx}{2 (-d)^{3/2}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{3/2}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{3/2}}-\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}-\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (b^2 n^2\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.775516, size = 432, normalized size = 0.85 \[ \frac{-\frac{2 b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{2 b n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{2 b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{3/2}}-\frac{2 b^2 n^2 \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{3/2}}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}-\frac{2 b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{2 b n \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{(-d)^{3/2}}+\frac{d \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{(-d)^{5/2}}}{4 \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.638, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ \left ( e{x}^{2}+d \right ) ^{2}}}\, 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{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, 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*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{\left (d + e x^{2}\right )^{2}}\, dx \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{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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