Integrand size = 18, antiderivative size = 13 \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\frac {\log \left (-n x+x^n\right )}{n} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1607, 528, 457, 78} \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\frac {\log \left (1-n x^{1-n}\right )}{n}+\log (x) \]
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Rule 78
Rule 457
Rule 528
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \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 {\text {Subst}\left (\int \frac {1-x}{x (1-n x)} \, dx,x,x^{1-n}\right )}{1-n} \\ & = \frac {\text {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*}
Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\frac {\log \left (-n x+x^n\right )}{n} \]
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Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
risch | \(\frac {\ln \left (-n x +x^{n}\right )}{n}\) | \(14\) |
norman | \(\frac {\ln \left (n x -{\mathrm e}^{n \ln \left (x \right )}\right )}{n}\) | \(17\) |
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Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\frac {\log \left (-n x + x^{n}\right )}{n} \]
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Time = 0.78 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\begin {cases} \frac {\log {\left (x - \frac {x^{n}}{n} \right )}}{n} & \text {for}\: n \neq 0 \\- x + \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\frac {\log \left (n x - x^{n}\right )}{n} \]
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\[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\int { -\frac {x^{n - 1} - 1}{n x - x^{n}} \,d x } \]
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Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx=\frac {\ln \left (n\,x-x^n\right )}{n} \]
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