Optimal. Leaf size=14 \[ -4+\frac {4 x \log (4)}{x+x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.64, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 27, 32}
\begin {gather*} \frac {4 \log (4)}{x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left ((4 \log (4)) \int \frac {1}{1+2 x+x^2} \, dx\right )\\ &=-\left ((4 \log (4)) \int \frac {1}{(1+x)^2} \, dx\right )\\ &=\frac {4 \log (4)}{1+x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 9, normalized size = 0.64 \begin {gather*} \frac {4 \log (4)}{1+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.24, size = 10, normalized size = 0.71
method | result | size |
gosper | \(\frac {8 \ln \left (2\right )}{x +1}\) | \(10\) |
default | \(\frac {8 \ln \left (2\right )}{x +1}\) | \(10\) |
norman | \(\frac {8 \ln \left (2\right )}{x +1}\) | \(10\) |
risch | \(\frac {8 \ln \left (2\right )}{x +1}\) | \(10\) |
meijerg | \(-\frac {8 \ln \left (2\right ) x}{x +1}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 9, normalized size = 0.64 \begin {gather*} \frac {8 \, \log \left (2\right )}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 9, normalized size = 0.64 \begin {gather*} \frac {8 \, \log \left (2\right )}{x + 1} \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 = 0.50 \begin {gather*} \frac {8 \log {\left (2 \right )}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 9, normalized size = 0.64 \begin {gather*} \frac {8 \, \log \left (2\right )}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 9, normalized size = 0.64 \begin {gather*} \frac {8\,\ln \left (2\right )}{x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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