Optimal. Leaf size=53 \[ -\frac{b^2 \log \left (b x^n+2\right )}{8 n}+\frac{1}{8} b^2 \log (x)+\frac{b x^{-n}}{4 n}-\frac{x^{-2 n}}{4 n} \]
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Rubi [A] time = 0.0233912, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 44} \[ -\frac{b^2 \log \left (b x^n+2\right )}{8 n}+\frac{1}{8} b^2 \log (x)+\frac{b x^{-n}}{4 n}-\frac{x^{-2 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-2 n}}{2+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 (2+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x^3}-\frac{b}{4 x^2}+\frac{b^2}{8 x}-\frac{b^3}{8 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-2 n}}{4 n}+\frac{b x^{-n}}{4 n}+\frac{1}{8} b^2 \log (x)-\frac{b^2 \log \left (2+b x^n\right )}{8 n}\\ \end{align*}
Mathematica [A] time = 0.0413474, size = 42, normalized size = 0.79 \[ -\frac{b^2 \log \left (b x^n+2\right )-b^2 n \log (x)+x^{-2 n} \left (2-2 b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 59, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{8}}-{\frac{1}{4\,n}}+{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{4\,n}} \right ) }-{\frac{{b}^{2}\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{8\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971497, size = 63, normalized size = 1.19 \begin{align*} \frac{1}{8} \, b^{2} \log \left (x\right ) - \frac{b^{2} \log \left (\frac{b x^{n} + 2}{b}\right )}{8 \, n} + \frac{b x^{n} - 1}{4 \, n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05591, size = 112, normalized size = 2.11 \begin{align*} \frac{b^{2} n x^{2 \, n} \log \left (x\right ) - b^{2} x^{2 \, n} \log \left (b x^{n} + 2\right ) + 2 \, b x^{n} - 2}{8 \, n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{x^{-2 \, n - 1}}{b x^{n} + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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