Optimal. Leaf size=13 \[ \frac{\log \left (x^n-n x\right )}{n} \]
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Rubi [A] time = 0.0455255, antiderivative size = 20, normalized size of antiderivative = 1.54, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1593, 514, 446, 72} \[ \frac{\log \left (1-n x^{1-n}\right )}{n}+\log (x) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 514
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{-1+x^{-1+n}}{-n x+x^n} \, dx &=\int \frac{x^{-n} \left (-1+x^{-1+n}\right )}{1-n x^{1-n}} \, dx\\ &=\int \frac{1-x^{1-n}}{x \left (1-n x^{1-n}\right )} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1-x}{x (1-n x)} \, dx,x,x^{1-n}\right )}{1-n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x}+\frac{1-n}{-1+n x}\right ) \, dx,x,x^{1-n}\right )}{1-n}\\ &=\log (x)+\frac{\log \left (1-n x^{1-n}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0229577, size = 20, normalized size = 1.54 \[ \frac{\log \left (1-n x^{1-n}\right )}{n}+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 17, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( nx-{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938639, size = 19, normalized size = 1.46 \begin{align*} \frac{\log \left (n x - x^{n}\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22595, size = 26, normalized size = 2. \begin{align*} \frac{\log \left (-n x + x^{n}\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.51136, size = 14, normalized size = 1.08 \begin{align*} \begin{cases} \frac{\log{\left (- n x + x^{n} \right )}}{n} & \text{for}\: n \neq 0 \\- x + \log{\left (x \right )} & \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{x^{n - 1} - 1}{n x - x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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