Optimal. Leaf size=23 \[ \frac {(a+b \log (x))^{1-n}}{b (1-n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30}
\begin {gather*} \frac {(a+b \log (x))^{1-n}}{b (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2339
Rubi steps
\begin {align*} \int \frac {(a+b \log (x))^{-n}}{x} \, dx &=\frac {\text {Subst}\left (\int x^{-n} \, dx,x,a+b \log (x)\right )}{b}\\ &=\frac {(a+b \log (x))^{1-n}}{b (1-n)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {(a+b \log (x))^{1-n}}{b (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 24, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {\left (a +b \ln \left (x \right )\right )^{1-n}}{b \left (1-n \right )}\) | \(24\) |
default | \(\frac {\left (a +b \ln \left (x \right )\right )^{1-n}}{b \left (1-n \right )}\) | \(24\) |
risch | \(-\frac {\left (a +b \ln \left (x \right )\right ) \left (a +b \ln \left (x \right )\right )^{-n}}{b \left (-1+n \right )}\) | \(27\) |
norman | \(\left (-\frac {\ln \left (x \right )}{-1+n}-\frac {a}{b \left (-1+n \right )}\right ) {\mathrm e}^{-n \ln \left (a +b \ln \left (x \right )\right )}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.85, size = 27, normalized size = 1.17 \begin {gather*} -\frac {b \log \left (x\right ) + a}{{\left (b n - b\right )} {\left (b \log \left (x\right ) + a\right )}^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (14) = 28\).
time = 5.60, size = 71, normalized size = 3.09 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge n = 1 \\a^{- n} \log {\left (x \right )} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + \log {\left (x \right )} \right )}}{b} & \text {for}\: n = 1 \\- \frac {a}{b n \left (a + b \log {\left (x \right )}\right )^{n} - b \left (a + b \log {\left (x \right )}\right )^{n}} - \frac {b \log {\left (x \right )}}{b n \left (a + b \log {\left (x \right )}\right )^{n} - b \left (a + b \log {\left (x \right )}\right )^{n}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 22, normalized size = 0.96 \begin {gather*} -\frac {{\left (b \log \left (x\right ) + a\right )}^{-n + 1}}{b {\left (n - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 22, normalized size = 0.96 \begin {gather*} -\frac {{\left (a+b\,\ln \left (x\right )\right )}^{1-n}}{b\,\left (n-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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