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 = {536, 209}
\begin {gather*} 3 \text {ArcTan}(x)-\sqrt {2} \text {ArcTan}\left (\frac {x}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 536
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 \begin {gather*} 3 \tan ^{-1}(x)-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 18, normalized size = 0.90
method | result | size |
default | \(3 \arctan \left (x \right )-\arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\) | \(18\) |
risch | \(3 \arctan \left (x \right )-\arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 17, normalized size = 0.85 \begin {gather*} -\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 3 \, \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 = 17, normalized size = 0.85 \begin {gather*} -\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 3 \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.95 \begin {gather*} 3 \operatorname {atan}{\left (x \right )} - \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.65, size = 17, normalized size = 0.85 \begin {gather*} -\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 3 \, \arctan \left (x\right ) \end {gather*}
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 \begin {gather*} 3\,\mathrm {atan}\left (x\right )-\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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