Optimal. Leaf size=26 \[ x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0075295, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 261} \[ x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4620
Rule 261
Rubi steps
\begin{align*} \int \cos ^{-1}(a x) \, dx &=x \cos ^{-1}(a x)+a \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{a}+x \cos ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0072581, size = 26, normalized size = 1. \[ x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 27, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax\arccos \left ( ax \right ) -\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.44063, size = 35, normalized size = 1.35 \begin{align*} \frac{a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.91552, size = 57, normalized size = 2.19 \begin{align*} \frac{a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.150393, size = 24, normalized size = 0.92 \begin{align*} \begin{cases} x \operatorname{acos}{\left (a x \right )} - \frac{\sqrt{- a^{2} x^{2} + 1}}{a} & \text{for}\: a \neq 0 \\\frac{\pi x}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15976, size = 35, normalized size = 1.35 \begin{align*} \frac{a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]