Optimal. Leaf size=13 \[ \frac {\log \left (x^n-n x\right )}{n} \]
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Rubi [A] time = 0.05, 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 72
Rule 446
Rule 514
Rule 1593
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*}
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Mathematica [A] time = 0.02, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {-1+x^{-1+n}}{-n x+x^n} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.12, size = 13, normalized size = 1.00 \[ \frac {\log \left (-n x + x^{n}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{n - 1} - 1}{n x - x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 14, normalized size = 1.08
method | result | size |
risch | \(\frac {\ln \left (-n x +x^{n}\right )}{n}\) | \(14\) |
norman | \(\frac {\ln \left (n x -{\mathrm e}^{n \ln \relax (x )}\right )}{n}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 14, normalized size = 1.08 \[ \frac {\log \left (n x - x^{n}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 14, normalized size = 1.08 \[ \frac {\ln \left (n\,x-x^n\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 14, normalized size = 1.08 \[ \begin {cases} \frac {\log {\left (x - \frac {x^{n}}{n} \right )}}{n} & \text {for}\: n \neq 0 \\- x + \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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