Optimal. Leaf size=13 \[ -\frac {2}{3} (1-x)^{3/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {26, 32} \begin {gather*} -\frac {2}{3} (1-x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 26
Rule 32
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{\sqrt {1+x}} \, dx &=\int \sqrt {1-x} \, dx\\ &=-\frac {2}{3} (1-x)^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 25, normalized size = 1.92 \begin {gather*} \frac {2 (x-1) \sqrt {1-x^2}}{3 \sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 0.06, size = 28, normalized size = 2.15 \begin {gather*} -\frac {2 \left (2 (x+1)-(x+1)^2\right )^{3/2}}{3 (x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.71, size = 19, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1} {\left (x - 1\right )}}{3 \, \sqrt {x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 15, normalized size = 1.15 \begin {gather*} -\frac {2}{3} \, {\left (-x + 1\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.00, size = 20, normalized size = 1.54 \begin {gather*} \frac {2 \left (x -1\right ) \sqrt {-x^{2}+1}}{3 \sqrt {x +1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.83, size = 12, normalized size = 0.92 \begin {gather*} \frac {2}{3} \, {\left (x - 1\right )} \sqrt {-x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.52, size = 20, normalized size = 1.54 \begin {gather*} \frac {\left (\frac {2\,x}{3}-\frac {2}{3}\right )\,\sqrt {1-x^2}}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________