Optimal. Leaf size=13 \[ 2 \log (4 x+1)-\tan ^{-1}(x) \]
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Rubi [A] time = 0.0347155, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1629, 204} \[ 2 \log (4 x+1)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1629
Rule 204
Rubi steps
\begin{align*} \int \frac{7-4 x+8 x^2}{(1+4 x) \left (1+x^2\right )} \, dx &=\int \left (\frac{8}{1+4 x}+\frac{1}{-1-x^2}\right ) \, dx\\ &=2 \log (1+4 x)+\int \frac{1}{-1-x^2} \, dx\\ &=-\tan ^{-1}(x)+2 \log (1+4 x)\\ \end{align*}
Mathematica [A] time = 0.0086843, size = 13, normalized size = 1. \[ 2 \log (4 x+1)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 14, normalized size = 1.1 \begin{align*} -\arctan \left ( x \right ) +2\,\ln \left ( 1+4\,x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67753, size = 18, normalized size = 1.38 \begin{align*} -\arctan \left (x\right ) + 2 \, \log \left (4 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78125, size = 39, normalized size = 3. \begin{align*} -\arctan \left (x\right ) + 2 \, \log \left (4 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.116184, size = 10, normalized size = 0.77 \begin{align*} 2 \log{\left (x + \frac{1}{4} \right )} - \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11811, size = 19, normalized size = 1.46 \begin{align*} -\arctan \left (x\right ) + 2 \, \log \left ({\left | 4 \, x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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