Optimal. Leaf size=15 \[ \frac{\log \left (2 x^n+3\right )}{2 n} \]
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Rubi [A] time = 0.0063023, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1593, 260} \[ \frac{\log \left (2 x^n+3\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{2 x+3 x^{1-n}} \, dx &=\int \frac{x^{-1+n}}{3+2 x^n} \, dx\\ &=\frac{\log \left (3+2 x^n\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0033049, size = 15, normalized size = 1. \[ \frac{\log \left (2 x^n+3\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 34, normalized size = 2.3 \begin{align*} -{\frac{\ln \left ( x \right ) }{2\,n}}+{\frac{\ln \left ( x \right ) }{2}}+{\frac{\ln \left ( 2\,x+3\,{{\rm e}^{ \left ( 1-n \right ) \ln \left ( x \right ) }} \right ) }{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04812, size = 15, normalized size = 1. \begin{align*} \frac{\log \left (x^{n} + \frac{3}{2}\right )}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.722041, size = 68, normalized size = 4.53 \begin{align*} \frac{{\left (n - 1\right )} \log \left (x\right ) + \log \left (3 \, x^{-n + 1} + 2 \, x\right )}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.64235, size = 22, normalized size = 1.47 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (\frac{2}{3} + x^{- n} \right )}}{2 n} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{5} & \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}{3 \, x^{-n + 1} + 2 \, x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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