Optimal. Leaf size=41 \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]
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Rubi [A] time = 0.311424, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {6694, 180} \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]
Antiderivative was successfully verified.
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Rule 6694
Rule 180
Rubi steps
\begin{align*} \int \frac{x}{\left (1+x^2\right ) \left (2+x^2\right ) \left (3+x^2\right ) \left (4+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1+x) (2+x) (3+x) (4+x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{6 (1+x)}-\frac{1}{2 (2+x)}+\frac{1}{2 (3+x)}-\frac{1}{6 (4+x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{12} \log \left (1+x^2\right )-\frac{1}{4} \log \left (2+x^2\right )+\frac{1}{4} \log \left (3+x^2\right )-\frac{1}{12} \log \left (4+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0083798, size = 41, normalized size = 1. \[ \frac{1}{12} \log \left (x^2+1\right )-\frac{1}{4} \log \left (x^2+2\right )+\frac{1}{4} \log \left (x^2+3\right )-\frac{1}{12} \log \left (x^2+4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 34, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{2}+1 \right ) }{12}}-{\frac{\ln \left ({x}^{2}+2 \right ) }{4}}+{\frac{\ln \left ({x}^{2}+3 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+4 \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.921042, size = 45, normalized size = 1.1 \begin{align*} -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17329, size = 105, normalized size = 2.56 \begin{align*} -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.161699, size = 32, normalized size = 0.78 \begin{align*} \frac{\log{\left (x^{2} + 1 \right )}}{12} - \frac{\log{\left (x^{2} + 2 \right )}}{4} + \frac{\log{\left (x^{2} + 3 \right )}}{4} - \frac{\log{\left (x^{2} + 4 \right )}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0602, size = 45, normalized size = 1.1 \begin{align*} -\frac{1}{12} \, \log \left (x^{2} + 4\right ) + \frac{1}{4} \, \log \left (x^{2} + 3\right ) - \frac{1}{4} \, \log \left (x^{2} + 2\right ) + \frac{1}{12} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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