Optimal. Leaf size=34 \[ \frac{a^x b^{-x}-a^{-x} b^x}{\log (a)-\log (b)}-2 x \]
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Rubi [A] time = 0.207676, antiderivative size = 41, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2287, 6742, 2194, 8} \[ -\frac{a^{-x} b^x}{\log (a)-\log (b)}+\frac{a^x b^{-x}}{\log (a)-\log (b)}-2 x \]
Antiderivative was successfully verified.
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Rule 2287
Rule 6742
Rule 2194
Rule 8
Rubi steps
\begin{align*} \int a^{-x} b^{-x} \left (a^x-b^x\right )^2 \, dx &=\int \left (a^x-b^x\right )^2 e^{-x (\log (a)+\log (b))} \, dx\\ &=\int \left (a^{2 x} e^{-x (\log (a)+\log (b))}-2 a^x b^x e^{-x (\log (a)+\log (b))}+b^{2 x} e^{-x (\log (a)+\log (b))}\right ) \, dx\\ &=-\left (2 \int a^x b^x e^{-x (\log (a)+\log (b))} \, dx\right )+\int a^{2 x} e^{-x (\log (a)+\log (b))} \, dx+\int b^{2 x} e^{-x (\log (a)+\log (b))} \, dx\\ &=-(2 \int 1 \, dx)+\int e^{-x (\log (a)-\log (b))} \, dx+\int e^{x (\log (a)-\log (b))} \, dx\\ &=-2 x+\frac{a^x b^{-x}}{\log (a)-\log (b)}-\frac{a^{-x} b^x}{\log (a)-\log (b)}\\ \end{align*}
Mathematica [A] time = 0.0499614, size = 46, normalized size = 1.35 \[ \frac{e^{x (\log (a)-\log (b))}}{\log (a)-\log (b)}+\frac{e^{x (\log (b)-\log (a))}}{\log (b)-\log (a)}-2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 65, normalized size = 1.9 \begin{align*}{\frac{1}{{{\rm e}^{x\ln \left ( a \right ) }}{{\rm e}^{x\ln \left ( b \right ) }}} \left ({\frac{ \left ({{\rm e}^{x\ln \left ( a \right ) }} \right ) ^{2}}{\ln \left ( a \right ) -\ln \left ( b \right ) }}-{\frac{ \left ({{\rm e}^{x\ln \left ( b \right ) }} \right ) ^{2}}{\ln \left ( a \right ) -\ln \left ( b \right ) }}-2\,x{{\rm e}^{x\ln \left ( a \right ) }}{{\rm e}^{x\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 = 1.82862, size = 113, normalized size = 3.32 \begin{align*} -\frac{2 \,{\left (x \log \left (a\right ) - x \log \left (b\right )\right )} a^{x} b^{x} - a^{2 \, x} + b^{2 \, x}}{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 [C] time = 1.30936, size = 589, normalized size = 17.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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