Optimal. Leaf size=26 \[ \frac{\log (b+x)}{a-b}-\frac{\log (a+x)}{a-b} \]
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Rubi [A] time = 0.0057296, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {36, 31} \[ \frac{\log (b+x)}{a-b}-\frac{\log (a+x)}{a-b} \]
Antiderivative was successfully verified.
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Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(a+x) (b+x)} \, dx &=\frac{\int \frac{1}{a+x} \, dx}{-a+b}-\frac{\int \frac{1}{b+x} \, dx}{-a+b}\\ &=-\frac{\log (a+x)}{a-b}+\frac{\log (b+x)}{a-b}\\ \end{align*}
Mathematica [A] time = 0.0065485, size = 19, normalized size = 0.73 \[ \frac{\log (b+x)-\log (a+x)}{a-b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 27, normalized size = 1. \begin{align*} -{\frac{\ln \left ( a+x \right ) }{a-b}}+{\frac{\ln \left ( b+x \right ) }{a-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928666, size = 35, normalized size = 1.35 \begin{align*} -\frac{\log \left (a + x\right )}{a - b} + \frac{\log \left (b + x\right )}{a - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8644, size = 49, normalized size = 1.88 \begin{align*} -\frac{\log \left (a + x\right ) - \log \left (b + x\right )}{a - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.197396, size = 80, normalized size = 3.08 \begin{align*} \frac{\log{\left (- \frac{a^{2}}{2 \left (a - b\right )} + \frac{a b}{a - b} + \frac{a}{2} - \frac{b^{2}}{2 \left (a - b\right )} + \frac{b}{2} + x \right )}}{a - b} - \frac{\log{\left (\frac{a^{2}}{2 \left (a - b\right )} - \frac{a b}{a - b} + \frac{a}{2} + \frac{b^{2}}{2 \left (a - b\right )} + \frac{b}{2} + x \right )}}{a - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05028, size = 38, normalized size = 1.46 \begin{align*} -\frac{\log \left ({\left | a + x \right |}\right )}{a - b} + \frac{\log \left ({\left | b + x \right |}\right )}{a - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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