Optimal. Leaf size=13 \[ \frac {\log \left (-n x+x^n\right )}{n} \]
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Rubi [A]
time = 0.03, 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 = {1607, 528, 457,
78} \begin {gather*} \frac {\log \left (1-n x^{1-n}\right )}{n}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 528
Rule 1607
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 {\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*}
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Mathematica [A]
time = 0.02, size = 13, normalized size = 1.00 \begin {gather*} \frac {\log \left (-n x+x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 14, normalized size = 1.08
method | result | size |
risch | \(\frac {\ln \left (-n x +x^{n}\right )}{n}\) | \(14\) |
norman | \(\frac {\ln \left (n x -{\mathrm e}^{n \ln \left (x \right )}\right )}{n}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.50, size = 14, normalized size = 1.08 \begin {gather*} \frac {\log \left (n x - x^{n}\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 13, normalized size = 1.00 \begin {gather*} \frac {\log \left (-n x + x^{n}\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.79, size = 14, normalized size = 1.08 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\ln \left (n\,x-x^n\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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