Optimal. Leaf size=509 \[ \frac {b n \text {Li}_2\left (-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {b n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {b n \log \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {b n \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\log \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt {e}}+\frac {\log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt {e}}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {b^2 n^2 \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}-\frac {b^2 n^2 \text {Li}_3\left (-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}}+\frac {b^2 n^2 \text {Li}_3\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 (-d)^{3/2} \sqrt {e}} \]
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Rubi [A] time = 0.61, antiderivative size = 509, normalized size of antiderivative = 1.00, 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 2317
Rule 2318
Rule 2330
Rule 2374
Rule 2391
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*}
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Mathematica [A] time = 0.78, size = 432, normalized size = 0.85 \[ \frac {-\frac {2 b n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac {2 b n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}-\frac {2 b n \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac {2 b n \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{(-d)^{3/2}}+\frac {d \log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{(-d)^{5/2}}+\frac {2 b^2 n^2 \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2}}-\frac {2 b^2 n^2 \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}+\frac {2 b^2 n^2 \text {Li}_3\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2}}-\frac {2 b^2 n^2 \text {Li}_3\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}}{4 \sqrt {e}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 27.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}{\left (e \,x^{2}+d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a^{2} {\left (\frac {x}{d e x^{2} + d^{2}} + \frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d}\right )} + \int \frac {b^{2} \log \relax (c)^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left (x^{n}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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