Optimal. Leaf size=26 \[ \sqrt{x^2+2 x}-\tan ^{-1}\left (\sqrt{x^2+2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0143339, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {685, 688, 203} \[ \sqrt{x^2+2 x}-\tan ^{-1}\left (\sqrt{x^2+2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 685
Rule 688
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{2 x+x^2}}{1+x} \, dx &=\sqrt{2 x+x^2}-\int \frac{1}{(1+x) \sqrt{2 x+x^2}} \, dx\\ &=\sqrt{2 x+x^2}-4 \operatorname{Subst}\left (\int \frac{1}{4+4 x^2} \, dx,x,\sqrt{2 x+x^2}\right )\\ &=\sqrt{2 x+x^2}-\tan ^{-1}\left (\sqrt{2 x+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0285005, size = 38, normalized size = 1.46 \[ \sqrt{x (x+2)} \left (1-\frac{2 \tan ^{-1}\left (\sqrt{\frac{x}{x+2}}\right )}{\sqrt{x} \sqrt{x+2}}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 21, normalized size = 0.8 \begin{align*} \sqrt{ \left ( 1+x \right ) ^{2}-1}+\arctan \left ({\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-1}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.72709, size = 23, normalized size = 0.88 \begin{align*} \sqrt{x^{2} + 2 \, x} + \arcsin \left (\frac{1}{{\left | x + 1 \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.09229, size = 73, normalized size = 2.81 \begin{align*} \sqrt{x^{2} + 2 \, x} - 2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (x + 2\right )}}{x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.3209, size = 36, normalized size = 1.38 \begin{align*} \sqrt{x^{2} + 2 \, x} - 2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]