Optimal. Leaf size=18 \[ \frac {a^x b^{-x}}{\log (a)-\log (b)} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 2194
Rule 2287
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*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 1.00 \[ \frac {a^x b^{-x}}{\log (a)-\log (b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 18, normalized size = 1.00 \[ \frac {a^{x}}{b^{x} {\left (\log \relax (a) - \log \relax (b)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 231, normalized size = 12.83 \[ 2 \, {\left (\frac {2 \, {\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) + \frac {1}{2} \, \pi x \mathrm {sgn}\relax (b)\right )}{{\left (\pi \mathrm {sgn}\relax (a) - \pi \mathrm {sgn}\relax (b)\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}^{2}} - \frac {{\left (\pi \mathrm {sgn}\relax (a) - \pi \mathrm {sgn}\relax (b)\right )} \sin \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) + \frac {1}{2} \, \pi x \mathrm {sgn}\relax (b)\right )}{{\left (\pi \mathrm {sgn}\relax (a) - \pi \mathrm {sgn}\relax (b)\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}\right )} - \frac {{\left (\frac {i e^{\left (\frac {1}{2} \, {\left (\pi {\left (\mathrm {sgn}\relax (a) - 1\right )} - \pi {\left (\mathrm {sgn}\relax (b) - 1\right )}\right )} i x\right )}}{\pi i \mathrm {sgn}\relax (a) - \pi i \mathrm {sgn}\relax (b) + 2 \, \log \left ({\left | a \right |}\right ) - 2 \, \log \left ({\left | b \right |}\right )} + \frac {i e^{\left (-\frac {1}{2} \, {\left (\pi {\left (\mathrm {sgn}\relax (a) - 1\right )} - \pi {\left (\mathrm {sgn}\relax (b) - 1\right )}\right )} i x\right )}}{\pi i \mathrm {sgn}\relax (a) - \pi i \mathrm {sgn}\relax (b) - 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right )}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}\right )}}{i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 1.06 \[ \frac {a^{x} b^{-x}}{\ln \relax (a )-\ln \relax (b )} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 18, normalized size = 1.00 \[ \frac {a^x}{b^x\,\left (\ln \relax (a)-\ln \relax (b)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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