Optimal. Leaf size=10 \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.002131, antiderivative size = 10, normalized size of antiderivative = 1., 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 = {41, 216} \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-3 x} \sqrt{2+3 x}} \, dx &=\int \frac{1}{\sqrt{4-9 x^2}} \, dx\\ &=\frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0043825, size = 10, normalized size = 1. \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 34, normalized size = 3.4 \begin{align*}{\frac{1}{3}\sqrt{ \left ( 2-3\,x \right ) \left ( 2+3\,x \right ) }\arcsin \left ({\frac{3\,x}{2}} \right ){\frac{1}{\sqrt{2-3\,x}}}{\frac{1}{\sqrt{2+3\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.62172, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.50983, size = 74, normalized size = 7.4 \begin{align*} -\frac{2}{3} \, \arctan \left (\frac{\sqrt{3 \, x + 2} \sqrt{-3 \, x + 2} - 2}{3 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.04167, size = 51, normalized size = 5.1 \begin{align*} \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{3} \sqrt{x + \frac{2}{3}}}{2} \right )}}{3} & \text{for}\: \frac{3 \left |{x + \frac{2}{3}}\right |}{4} > 1 \\\frac{2 \operatorname{asin}{\left (\frac{\sqrt{3} \sqrt{x + \frac{2}{3}}}{2} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1356, size = 16, normalized size = 1.6 \begin{align*} \frac{2}{3} \, \arcsin \left (\frac{1}{2} \, \sqrt{3 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]