Optimal. Leaf size=22 \[ \frac{\log (x)}{2}-\frac{\log \left (b x^n+2\right )}{2 n} \]
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Rubi [A] time = 0.00948, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{2}-\frac{\log \left (b x^n+2\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \left (2+b x^n\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (2+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^n\right )}{2 n}-\frac{b \operatorname{Subst}\left (\int \frac{1}{2+b x} \, dx,x,x^n\right )}{2 n}\\ &=\frac{\log (x)}{2}-\frac{\log \left (2+b x^n\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0051662, size = 22, normalized size = 1. \[ \frac{n \log (x)-\log \left (b x^n+2\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 24, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({x}^{n} \right ) }{2\,n}}-{\frac{\ln \left ( 2+b{x}^{n} \right ) }{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953897, size = 31, normalized size = 1.41 \begin{align*} -\frac{\log \left (b x^{n} + 2\right )}{2 \, n} + \frac{\log \left (x^{n}\right )}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03745, size = 47, normalized size = 2.14 \begin{align*} \frac{n \log \left (x\right ) - \log \left (b x^{n} + 2\right )}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.550214, size = 31, normalized size = 1.41 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{b + 2} & \text{for}\: n = 0 \\\frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{2} - \frac{\log{\left (x^{n} + \frac{2}{b} \right )}}{2 n} & \text{otherwise} \end{cases} \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{1}{{\left (b x^{n} + 2\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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