Optimal. Leaf size=22 \[ -2 \log \left (x^2+1\right )-\frac{4}{x}+4 \log (x)-4 \tan ^{-1}(x) \]
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Rubi [A] time = 0.0189116, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {801, 635, 203, 260} \[ -2 \log \left (x^2+1\right )-\frac{4}{x}+4 \log (x)-4 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{4+4 x}{x^2 \left (1+x^2\right )} \, dx &=\int \left (\frac{4}{x^2}+\frac{4}{x}-\frac{4 (1+x)}{1+x^2}\right ) \, dx\\ &=-\frac{4}{x}+4 \log (x)-4 \int \frac{1+x}{1+x^2} \, dx\\ &=-\frac{4}{x}+4 \log (x)-4 \int \frac{1}{1+x^2} \, dx-4 \int \frac{x}{1+x^2} \, dx\\ &=-\frac{4}{x}-4 \tan ^{-1}(x)+4 \log (x)-2 \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0054922, size = 24, normalized size = 1.09 \[ 4 \left (-\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{x}+\log (x)-\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 23, normalized size = 1.1 \begin{align*} -4\,{x}^{-1}-4\,\arctan \left ( x \right ) +4\,\ln \left ( x \right ) -2\,\ln \left ({x}^{2}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48236, size = 30, normalized size = 1.36 \begin{align*} -\frac{4}{x} - 4 \, \arctan \left (x\right ) - 2 \, \log \left (x^{2} + 1\right ) + 4 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.999919, size = 76, normalized size = 3.45 \begin{align*} -\frac{2 \,{\left (2 \, x \arctan \left (x\right ) + x \log \left (x^{2} + 1\right ) - 2 \, x \log \left (x\right ) + 2\right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.117294, size = 20, normalized size = 0.91 \begin{align*} 4 \log{\left (x \right )} - 2 \log{\left (x^{2} + 1 \right )} - 4 \operatorname{atan}{\left (x \right )} - \frac{4}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1062, size = 31, normalized size = 1.41 \begin{align*} -\frac{4}{x} - 4 \, \arctan \left (x\right ) - 2 \, \log \left (x^{2} + 1\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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