Optimal. Leaf size=23 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0026651, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {195, 216} \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-x^2} \, dx &=\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0051649, size = 20, normalized size = 0.87 \[ \frac{1}{2} \left (\sqrt{1-x^2} x+\sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 18, normalized size = 0.8 \begin{align*}{\frac{\arcsin \left ( x \right ) }{2}}+{\frac{x}{2}\sqrt{-{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.67506, size = 23, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.47047, size = 74, normalized size = 3.22 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1} x - \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.184459, size = 15, normalized size = 0.65 \begin{align*} \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12368, size = 23, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]