Optimal. Leaf size=31 \[ a x-\frac{b \sqrt{1-c^2 x^2}}{c}+b x \cos ^{-1}(c x) \]
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Rubi [A] time = 0.019312, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4620, 261} \[ a x-\frac{b \sqrt{1-c^2 x^2}}{c}+b x \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 4620
Rule 261
Rubi steps
\begin{align*} \int \left (a+b \cos ^{-1}(c x)\right ) \, dx &=a x+b \int \cos ^{-1}(c x) \, dx\\ &=a x+b x \cos ^{-1}(c x)+(b c) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx\\ &=a x-\frac{b \sqrt{1-c^2 x^2}}{c}+b x \cos ^{-1}(c x)\\ \end{align*}
Mathematica [A] time = 0.0121239, size = 31, normalized size = 1. \[ a x-\frac{b \sqrt{1-c^2 x^2}}{c}+b x \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 1. \begin{align*} ax+{\frac{b}{c} \left ( cx\arccos \left ( cx \right ) -\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43553, size = 42, normalized size = 1.35 \begin{align*} a x + \frac{{\left (c x \arccos \left (c x\right ) - \sqrt{-c^{2} x^{2} + 1}\right )} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49453, size = 73, normalized size = 2.35 \begin{align*} \frac{b c x \arccos \left (c x\right ) + a c x - \sqrt{-c^{2} x^{2} + 1} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.153665, size = 29, normalized size = 0.94 \begin{align*} a x + b \left (\begin{cases} x \operatorname{acos}{\left (c x \right )} - \frac{\sqrt{- c^{2} x^{2} + 1}}{c} & \text{for}\: c \neq 0 \\\frac{\pi x}{2} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12345, size = 42, normalized size = 1.35 \begin{align*} a x + \frac{{\left (c x \arccos \left (c x\right ) - \sqrt{-c^{2} x^{2} + 1}\right )} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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