Optimal. Leaf size=12 \[ \tan ^{-1}\left (2 \sqrt {x+x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {702, 209}
\begin {gather*} \text {ArcTan}\left (2 \sqrt {x^2+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 702
Rubi steps
\begin {align*} \int \frac {1}{(1+2 x) \sqrt {x+x^2}} \, dx &=4 \text {Subst}\left (\int \frac {1}{2+8 x^2} \, dx,x,\sqrt {x+x^2}\right )\\ &=\tan ^{-1}\left (2 \sqrt {x+x^2}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(12)=24\).
time = 0.04, size = 43, normalized size = 3.58 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {1+x} \tan ^{-1}\left (1+2 x-2 \sqrt {x} \sqrt {1+x}\right )}{\sqrt {x (1+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 15, normalized size = 1.25
method | result | size |
default | \(-\arctan \left (\frac {1}{\sqrt {4 \left (x +\frac {1}{2}\right )^{2}-1}}\right )\) | \(15\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{2}+x}}{2 x +1}\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 11, normalized size = 0.92 \begin {gather*} -\arcsin \left (\frac {1}{{\left | 2 \, x + 1 \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 17, normalized size = 1.42 \begin {gather*} 2 \, \arctan \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x \left (x + 1\right )} \left (2 x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.88, size = 17, normalized size = 1.42 \begin {gather*} 2 \, \arctan \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {1}{\left (2\,x+1\right )\,\sqrt {x^2+x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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