Optimal. Leaf size=301 \[ -\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}+1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}+\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{\sqrt{d-e x} \sqrt{d+e x}} \]
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Rubi [A] time = 0.604864, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2342, 266, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}+1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}+\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{\sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 2342
Rule 266
Rule 63
Rule 208
Rule 2348
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x \sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \int \frac{a+b \log \left (c x^n\right )}{x \sqrt{1-\frac{e^2 x^2}{d^2}}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{x} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\sqrt{1-\frac{e^2 x}{d^2}}\right )}{x} \, dx,x,x^2\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}(x)}{-1+x^2} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{1-x} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{\sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{\sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{\sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{Li}_2\left (1-\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 1.75835, size = 310, normalized size = 1.03 \[ \frac{b n \sqrt{e^2 x^2-d^2} \left (\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \left (-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{1-\frac{e^2 x^2}{d^2}}\right )+\log ^2\left (\frac{e^2 x^2}{d^2}\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}}+1\right )\right )-4 \log \left (\frac{1}{2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}}+1\right )\right ) \log \left (\frac{e^2 x^2}{d^2}\right )\right )}{\sqrt{e^2 x^2-d^2}}-\frac{4 \left (2 \log (x)-\log \left (\frac{e^2 x^2}{d^2}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{e^2 x^2-d^2}}{\sqrt{-d^2}}\right )}{\sqrt{-d^2}}\right )}{8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d}+\frac{\log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.628, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{x}{\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{ex+d}}}}\, 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{\sqrt{e x + d} \sqrt{-e x + d} b \log \left (c x^{n}\right ) + \sqrt{e x + d} \sqrt{-e x + d} a}{e^{2} x^{3} - d^{2} x}, 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{a + b \log{\left (c x^{n} \right )}}{x \sqrt{d - e x} \sqrt{d + e x}}\, 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{b \log \left (c x^{n}\right ) + a}{\sqrt{e x + d} \sqrt{-e x + d} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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