Optimal. Leaf size=23 \[ \log \left (\frac {e^{4 e^9} \left ((-1-2 x)^2+x\right )^4}{x^4}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1608, 1642}
\begin {gather*} -4 \log (x)+4 \log (x+1)+4 \log (4 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 1608
Rule 1642
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+16 x^2}{x \left (1+5 x+4 x^2\right )} \, dx\\ &=\int \left (-\frac {4}{x}+\frac {4}{1+x}+\frac {16}{1+4 x}\right ) \, dx\\ &=-4 \log (x)+4 \log (1+x)+4 \log (1+4 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.78 \begin {gather*} -4 \log (x)+4 \log \left (1+5 x+4 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 20, normalized size = 0.87
method | result | size |
risch | \(-4 \ln \left (x \right )+4 \ln \left (4 x^{2}+5 x +1\right )\) | \(19\) |
default | \(4 \ln \left (x +1\right )-4 \ln \left (x \right )+4 \ln \left (4 x +1\right )\) | \(20\) |
norman | \(4 \ln \left (x +1\right )-4 \ln \left (x \right )+4 \ln \left (4 x +1\right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 19, normalized size = 0.83 \begin {gather*} 4 \, \log \left (4 \, x + 1\right ) + 4 \, \log \left (x + 1\right ) - 4 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 18, normalized size = 0.78 \begin {gather*} 4 \, \log \left (4 \, x^{2} + 5 \, x + 1\right ) - 4 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 17, normalized size = 0.74 \begin {gather*} - 4 \log {\left (x \right )} + 4 \log {\left (4 x^{2} + 5 x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 22, normalized size = 0.96 \begin {gather*} 4 \, \log \left ({\left | 4 \, x + 1 \right |}\right ) + 4 \, \log \left ({\left | x + 1 \right |}\right ) - 4 \, \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 = 4.12, size = 16, normalized size = 0.70 \begin {gather*} 4\,\ln \left (x^2+\frac {5\,x}{4}+\frac {1}{4}\right )-4\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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