Optimal. Leaf size=8 \[ 2 \tan ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {661, 65, 209}
\begin {gather*} 2 \text {ArcTan}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 209
Rule 661
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{x+x^2} \, dx &=\int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(2 \arctan \left (\sqrt {x}\right )\) | \(7\) |
default | \(2 \arctan \left (\sqrt {x}\right )\) | \(7\) |
meijerg | \(2 \arctan \left (\sqrt {x}\right )\) | \(7\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x}+x -1}{1+x}\right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.17, size = 7, normalized size = 0.88 \begin {gather*} 2 \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.61, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.14, size = 6, normalized size = 0.75 \begin {gather*} 2\,\mathrm {atan}\left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________