Optimal. Leaf size=25 \[ x-\frac {3}{2} \tan ^{-1}\left (\frac {1+x}{2}\right )-\log \left (5+2 x+x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {717, 648, 632,
210, 642} \begin {gather*} -\frac {3}{2} \text {ArcTan}\left (\frac {x+1}{2}\right )-\log \left (x^2+2 x+5\right )+x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 717
Rubi steps
\begin {align*} \int \frac {x^2}{5+2 x+x^2} \, dx &=x+\int \frac {-5-2 x}{5+2 x+x^2} \, dx\\ &=x-3 \int \frac {1}{5+2 x+x^2} \, dx-\int \frac {2+2 x}{5+2 x+x^2} \, dx\\ &=x-\log \left (5+2 x+x^2\right )+6 \text {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,2+2 x\right )\\ &=x-\frac {3}{2} \tan ^{-1}\left (\frac {1+x}{2}\right )-\log \left (5+2 x+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} x-\frac {3}{2} \tan ^{-1}\left (\frac {1+x}{2}\right )-\log \left (5+2 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 22, normalized size = 0.88
method | result | size |
default | \(x -\frac {3 \arctan \left (\frac {1}{2}+\frac {x}{2}\right )}{2}-\ln \left (x^{2}+2 x +5\right )\) | \(22\) |
risch | \(x -\frac {3 \arctan \left (\frac {1}{2}+\frac {x}{2}\right )}{2}-\ln \left (x^{2}+2 x +5\right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.37, size = 21, normalized size = 0.84 \begin {gather*} x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 21, normalized size = 0.84 \begin {gather*} x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.03, size = 22, normalized size = 0.88 \begin {gather*} x - \log {\left (x^{2} + 2 x + 5 \right )} - \frac {3 \operatorname {atan}{\left (\frac {x}{2} + \frac {1}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.82, size = 21, normalized size = 0.84 \begin {gather*} x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 21, normalized size = 0.84 \begin {gather*} x-\ln \left (x^2+2\,x+5\right )-\frac {3\,\mathrm {atan}\left (\frac {x}{2}+\frac {1}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________