Optimal. Leaf size=20 \[ 3 \tan ^{-1}(x)-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 203} \[ 3 \tan ^{-1}(x)-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 522
Rubi steps
\begin {align*} \int \frac {4+x^2}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx &=-\left (2 \int \frac {1}{2+x^2} \, dx\right )+3 \int \frac {1}{1+x^2} \, dx\\ &=3 \tan ^{-1}(x)-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ 3 \tan ^{-1}(x)-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 17, normalized size = 0.85 \[ -\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 3 \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 17, normalized size = 0.85 \[ -\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 3 \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 18, normalized size = 0.90 \[ 3 \arctan \relax (x )-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.40, size = 17, normalized size = 0.85 \[ -\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 3 \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 17, normalized size = 0.85 \[ 3\,\mathrm {atan}\relax (x)-\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 19, normalized size = 0.95 \[ 3 \operatorname {atan}{\relax (x )} - \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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