Optimal. Leaf size=35 \[ -\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x \]
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Rubi [A] time = 0.0468431, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 4678, 8} \[ -\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x \]
Antiderivative was successfully verified.
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Rule 4620
Rule 4678
Rule 8
Rubi steps
\begin{align*} \int \cos ^{-1}(a x)^2 \, dx &=x \cos ^{-1}(a x)^2+(2 a) \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 \int 1 \, dx\\ &=-2 x-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2\\ \end{align*}
Mathematica [A] time = 0.0161168, size = 35, normalized size = 1. \[ -\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 37, normalized size = 1.1 \begin{align*}{\frac{1}{a} \left ( ax \left ( \arccos \left ( ax \right ) \right ) ^{2}-2\,ax-2\,\arccos \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43443, size = 45, normalized size = 1.29 \begin{align*} x \arccos \left (a x\right )^{2} - 2 \, x - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87394, size = 89, normalized size = 2.54 \begin{align*} \frac{a x \arccos \left (a x\right )^{2} - 2 \, a x - 2 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.238852, size = 37, normalized size = 1.06 \begin{align*} \begin{cases} x \operatorname{acos}^{2}{\left (a x \right )} - 2 x - \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\\frac{\pi ^{2} x}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13932, size = 45, normalized size = 1.29 \begin{align*} x \arccos \left (a x\right )^{2} - 2 \, x - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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