Optimal. Leaf size=23 \[ \frac{(a+b \log (x))^{1-n}}{b (1-n)} \]
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Rubi [A] time = 0.0324734, antiderivative size = 23, normalized size of antiderivative = 1., 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 = {2302, 30} \[ \frac{(a+b \log (x))^{1-n}}{b (1-n)} \]
Antiderivative was successfully verified.
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Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{(a+b \log (x))^{-n}}{x} \, dx &=\frac{\operatorname{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*}
Mathematica [A] time = 0.011729, size = 23, normalized size = 1. \[ \frac{(a+b \log (x))^{1-n}}{b (1-n)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 24, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\ln \left ( x \right ) \right ) ^{1-n}}{b \left ( 1-n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44858, size = 62, normalized size = 2.7 \begin{align*} -\frac{b \log \left (x\right ) + a}{{\left (b n - b\right )}{\left (b \log \left (x\right ) + a\right )}^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.2145, size = 71, normalized size = 3.09 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0971, size = 30, normalized size = 1.3 \begin{align*} -\frac{{\left (b \log \left (x\right ) + a\right )}^{-n + 1}}{b{\left (n - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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