Optimal. Leaf size=21 \[ \frac {1}{6} \log \left (1+x^2\right )-\frac {1}{6} \log \left (4+x^2\right ) \]
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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 = {455, 36, 31}
\begin {gather*} \frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 455
Rubi steps
\begin {align*} \int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) (4+x)} \, dx,x,x^2\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-\frac {1}{6} \text {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 \begin {gather*} \frac {1}{6} \log \left (1+x^2\right )-\frac {1}{6} \log \left (4+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 18, normalized size = 0.86
method | result | size |
default | \(\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6}\) | \(18\) |
norman | \(\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6}\) | \(18\) |
risch | \(\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.86, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x^{2} + 1 \right )}}{6} - \frac {\log {\left (x^{2} + 4 \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.88, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 17, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {3\,x^2}{5\,x^2+8}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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