Optimal. Leaf size=15 \[ \frac{\log \left (a x^n+b\right )}{a n} \]
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Rubi [A] time = 0.0077497, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 260} \[ \frac{\log \left (a x^n+b\right )}{a n} \]
Antiderivative was successfully verified.
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Rule 263
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^{-n}\right )} \, dx &=\int \frac{x^{-1+n}}{b+a x^n} \, dx\\ &=\frac{\log \left (b+a x^n\right )}{a n}\\ \end{align*}
Mathematica [A] time = 0.004008, size = 15, normalized size = 1. \[ \frac{\log \left (a x^n+b\right )}{a n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 16, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( b+a{x}^{n} \right ) }{an}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.977192, size = 43, normalized size = 2.87 \begin{align*} \frac{\log \left (a + \frac{b}{x^{n}}\right )}{a n} - \frac{\log \left (\frac{1}{x^{n}}\right )}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27793, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (a x^{n} + b\right )}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.692164, size = 39, normalized size = 2.6 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{x^{n}}{b n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a} + \frac{\log{\left (\frac{a}{b} + x^{- n} \right )}}{a n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{n}}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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