Optimal. Leaf size=9 \[ \log (x)-\log (1+x) \]
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Rubi [A]
time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {36, 29, 31}
\begin {gather*} \log (x)-\log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {1}{x (1+x)} \, dx &=\int \frac {1}{x} \, dx-\int \frac {1}{1+x} \, dx\\ &=\log (x)-\log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \log (x)-\log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 10, normalized size = 1.11
method | result | size |
default | \(\ln \left (x \right )-\ln \left (1+x \right )\) | \(10\) |
norman | \(\ln \left (x \right )-\ln \left (1+x \right )\) | \(10\) |
meijerg | \(\ln \left (x \right )-\ln \left (1+x \right )\) | \(10\) |
risch | \(\ln \left (x \right )-\ln \left (1+x \right )\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.72, size = 9, normalized size = 1.00 \begin {gather*} -\log \left (x + 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 9, normalized size = 1.00 \begin {gather*} -\log \left (x + 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 7, normalized size = 0.78 \begin {gather*} \log {\left (x \right )} - \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 11, normalized size = 1.22 \begin {gather*} -\log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 8, normalized size = 0.89 \begin {gather*} -\ln \left (\frac {1}{x}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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