Optimal. Leaf size=250 \[ -\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{e} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{e} \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.139033, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {2327, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{e} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{e} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 2327
Rule 2325
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{\sqrt{d+e x^2}} \, dx &=\frac{\sqrt{1+\frac{e x^2}{d}} \int \frac{a+b \log \left (c x^n\right )}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{\sqrt{d+e x^2}}\\ &=\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}-\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{\sqrt{e} \sqrt{d+e x^2}}\\ &=\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}-\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{e} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{\left (2 b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{e} \sqrt{d+e x^2}}\\ &=\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{\sqrt{e} \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{e} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [A] time = 0.542951, size = 186, normalized size = 0.74 \[ \frac{b n \sqrt{\frac{e x^2}{d}+1} \left (\text{PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (x \sqrt{\frac{e}{d}}\right )}\right )+2 \log (x) \log \left (\sqrt{\frac{e x^2}{d}+1}+x \sqrt{\frac{e}{d}}\right )-\sinh ^{-1}\left (x \sqrt{\frac{e}{d}}\right )^2-2 \sinh ^{-1}\left (x \sqrt{\frac{e}{d}}\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (x \sqrt{\frac{e}{d}}\right )}\right )\right )}{2 \sqrt{\frac{e}{d}} \sqrt{d+e x^2}}+\frac{\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt{e}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.402, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c{x}^{n} \right ) ){\frac{1}{\sqrt{e{x}^{2}+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^{2} + d} b \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a}{e x^{2} + d}, 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 )}}{\sqrt{d + e x^{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{b \log \left (c x^{n}\right ) + a}{\sqrt{e x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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