Optimal. Leaf size=26 \[ \frac {\log (b+x)}{a-b}-\frac {\log (a+x)}{a-b} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 31
Rule 36
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*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 0.73 \[ \frac {\log (b+x)-\log (a+x)}{a-b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 20, normalized size = 0.77 \[ -\frac {\log \left (a + x\right ) - \log \left (b + x\right )}{a - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 28, normalized size = 1.08 \[ -\frac {\log \left ({\left | a + x \right |}\right )}{a - b} + \frac {\log \left ({\left | b + x \right |}\right )}{a - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 1.04 \[ -\frac {\ln \left (a +x \right )}{a -b}+\frac {\ln \left (b +x \right )}{a -b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 26, normalized size = 1.00 \[ -\frac {\log \left (a + x\right )}{a - b} + \frac {\log \left (b + x\right )}{a - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 18, normalized size = 0.69 \[ \frac {\ln \left (\frac {b+x}{a+x}\right )}{a-b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.23, size = 80, normalized size = 3.08 \[ \frac {\log {\left (- \frac {a^{2}}{2 \left (a - b\right )} + \frac {a b}{a - b} + \frac {a}{2} - \frac {b^{2}}{2 \left (a - b\right )} + \frac {b}{2} + x \right )}}{a - b} - \frac {\log {\left (\frac {a^{2}}{2 \left (a - b\right )} - \frac {a b}{a - b} + \frac {a}{2} + \frac {b^{2}}{2 \left (a - b\right )} + \frac {b}{2} + x \right )}}{a - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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