Optimal. Leaf size=11 \[ \log (x)+\frac {1}{2} \tan ^{-1}(2 x) \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1593, 1802, 203} \[ \log (x)+\frac {1}{2} \tan ^{-1}(2 x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1593
Rule 1802
Rubi steps
\begin {align*} \int \frac {1+x+4 x^2}{x+4 x^3} \, dx &=\int \frac {1+x+4 x^2}{x \left (1+4 x^2\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {1}{1+4 x^2}\right ) \, dx\\ &=\log (x)+\int \frac {1}{1+4 x^2} \, dx\\ &=\frac {1}{2} \tan ^{-1}(2 x)+\log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 11, normalized size = 1.00 \[ \log (x)+\frac {1}{2} \tan ^{-1}(2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 9, normalized size = 0.82 \[ \frac {1}{2} \, \arctan \left (2 \, x\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 10, normalized size = 0.91 \[ \frac {1}{2} \, \arctan \left (2 \, x\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 10, normalized size = 0.91 \[ \frac {\arctan \left (2 x \right )}{2}+\ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.61, size = 9, normalized size = 0.82 \[ \frac {1}{2} \, \arctan \left (2 \, x\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 17, normalized size = 1.55 \[ \ln \relax (x)-\frac {\mathrm {atan}\left (\frac {17}{32\,\left (\frac {x}{16}-\frac {1}{8}\right )}+4\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 8, normalized size = 0.73 \[ \log {\relax (x )} + \frac {\operatorname {atan}{\left (2 x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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