Optimal. Leaf size=21 \[ \frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {444, 36, 31} \[ \frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) (4+x)} \, dx,x,x^2\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{4+x} \, dx,x,x^2\right )\\ &=\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{6} \log \left (4+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 17, normalized size = 0.81 \[ -\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 17, normalized size = 0.81 \[ -\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 18, normalized size = 0.86 \[ \frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 17, normalized size = 0.81 \[ -\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 17, normalized size = 0.81 \[ \frac {\mathrm {atanh}\left (\frac {3\,x^2}{5\,x^2+8}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 15, normalized size = 0.71 \[ \frac {\log {\left (x^{2} + 1 \right )}}{6} - \frac {\log {\left (x^{2} + 4 \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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