Optimal. Leaf size=51 \[ -\frac {1}{12} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{6} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A]
time = 0.19, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6857, 209}
\begin {gather*} -\frac {1}{12} \text {ArcTan}\left (\frac {x}{2}\right )+\frac {\text {ArcTan}(x)}{6}-\frac {\text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {\text {ArcTan}\left (\frac {x}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 6857
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^2\right ) \left (2+x^2\right ) \left (3+x^2\right ) \left (4+x^2\right )} \, dx &=\int \left (\frac {1}{6 \left (1+x^2\right )}-\frac {1}{2 \left (2+x^2\right )}+\frac {1}{2 \left (3+x^2\right )}-\frac {1}{6 \left (4+x^2\right )}\right ) \, dx\\ &=\frac {1}{6} \int \frac {1}{1+x^2} \, dx-\frac {1}{6} \int \frac {1}{4+x^2} \, dx-\frac {1}{2} \int \frac {1}{2+x^2} \, dx+\frac {1}{2} \int \frac {1}{3+x^2} \, dx\\ &=-\frac {1}{12} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{6} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.92 \begin {gather*} \frac {1}{12} \left (-\tan ^{-1}\left (\frac {x}{2}\right )+2 \tan ^{-1}(x)-3 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 36, normalized size = 0.71
method | result | size |
default | \(-\frac {\arctan \left (\frac {x}{2}\right )}{12}+\frac {\arctan \left (x \right )}{6}-\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{4}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{6}\) | \(36\) |
risch | \(-\frac {\arctan \left (\frac {x}{2}\right )}{12}+\frac {\arctan \left (x \right )}{6}-\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{4}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{6}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.81, size = 35, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{12} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{6} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 35, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{12} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{6} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 44, normalized size = 0.86 \begin {gather*} - \frac {\operatorname {atan}{\left (\frac {x}{2} \right )}}{12} + \frac {\operatorname {atan}{\left (x \right )}}{6} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{4} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 35, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{12} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{6} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 35, normalized size = 0.69 \begin {gather*} \frac {\mathrm {atan}\left (x\right )}{6}-\frac {\mathrm {atan}\left (\frac {x}{2}\right )}{12}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{4}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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