Optimal. Leaf size=13 \[ 2 \log (4 x+1)-\tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1629, 204} \begin {gather*} 2 \log (4 x+1)-\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 1629
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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} 2 \log (4 x+1)-\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {7-4 x+8 x^2}{(1+4 x) \left (1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.18, size = 13, normalized size = 1.00 \begin {gather*} -\arctan \relax (x) + 2 \, \log \left (4 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 14, normalized size = 1.08 \begin {gather*} -\arctan \relax (x) + 2 \, \log \left ({\left | 4 \, x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 14, normalized size = 1.08 \begin {gather*} -\arctan \relax (x )+2 \ln \left (4 x +1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 13, normalized size = 1.00 \begin {gather*} -\arctan \relax (x) + 2 \, \log \left (4 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 19, normalized size = 1.46 \begin {gather*} \mathrm {atan}\left (\frac {4\,x+1}{x-4}\right )+2\,\ln \left (x+\frac {1}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 10, normalized size = 0.77 \begin {gather*} 2 \log {\left (x + \frac {1}{4} \right )} - \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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