Optimal. Leaf size=18 \[ \frac{a^x b^{-x}}{\log (a)-\log (b)} \]
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Rubi [A] time = 0.0211684, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2287, 2194} \[ \frac{a^x b^{-x}}{\log (a)-\log (b)} \]
Antiderivative was successfully verified.
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Rule 2287
Rule 2194
Rubi steps
\begin{align*} \int a^x b^{-x} \, dx &=\int e^{x (\log (a)-\log (b))} \, dx\\ &=\frac{a^x b^{-x}}{\log (a)-\log (b)}\\ \end{align*}
Mathematica [A] time = 0.0096114, size = 18, normalized size = 1. \[ \frac{a^x b^{-x}}{\log (a)-\log (b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 1.1 \begin{align*}{\frac{{a}^{x}}{{b}^{x} \left ( \ln \left ( a \right ) -\ln \left ( b \right ) \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 = 1.28415, size = 39, normalized size = 2.17 \begin{align*} \frac{a^{x}}{b^{x}{\left (\log \left (a\right ) - \log \left (b\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.49597, size = 312, normalized size = 17.33 \begin{align*} 2 \,{\left (\frac{2 \,{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )} \cos \left (-\frac{1}{2} \, \pi x \mathrm{sgn}\left (a\right ) + \frac{1}{2} \, \pi x \mathrm{sgn}\left (b\right )\right )}{{\left (\pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )}^{2} + 4 \,{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}^{2}} - \frac{{\left (\pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )} \sin \left (-\frac{1}{2} \, \pi x \mathrm{sgn}\left (a\right ) + \frac{1}{2} \, \pi x \mathrm{sgn}\left (b\right )\right )}{{\left (\pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )}^{2} + 4 \,{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}\right )} - \frac{{\left (\frac{i e^{\left (\frac{1}{2} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} - \pi{\left (\mathrm{sgn}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i \mathrm{sgn}\left (a\right ) - \pi i \mathrm{sgn}\left (b\right ) + 2 \, \log \left ({\left | a \right |}\right ) - 2 \, \log \left ({\left | b \right |}\right )} + \frac{i e^{\left (-\frac{1}{2} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} - \pi{\left (\mathrm{sgn}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i \mathrm{sgn}\left (a\right ) - \pi i \mathrm{sgn}\left (b\right ) - 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right )}\right )} e^{\left (x{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}\right )}}{i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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