Optimal. Leaf size=14 \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {620, 206} \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x+x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x+x^2}}\right )\\ &=2 \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right )\\ \end {align*}
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Mathematica [B] time = 0.01, size = 29, normalized size = 2.07 \[ \frac {2 \sqrt {x} \sqrt {x+1} \sinh ^{-1}\left (\sqrt {x}\right )}{\sqrt {x (x+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 17, normalized size = 1.21 \[ -\log \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.02, size = 18, normalized size = 1.29 \[ -\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 0.86 \[ \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 15, normalized size = 1.07 \[ \log \left (2 \, x + 2 \, \sqrt {x^{2} + x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 11, normalized size = 0.79 \[ \ln \left (x+\sqrt {x\,\left (x+1\right )}+\frac {1}{2}\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{2} + x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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