Optimal. Leaf size=47 \[ -\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{c}+x \left (a+b \cos ^{-1}(c x)\right )^2-2 b^2 x \]
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Rubi [A] time = 0.0578785, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4620, 4678, 8} \[ -\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{c}+x \left (a+b \cos ^{-1}(c x)\right )^2-2 b^2 x \]
Antiderivative was successfully verified.
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Rule 4620
Rule 4678
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \cos ^{-1}(c x)\right )^2 \, dx &=x \left (a+b \cos ^{-1}(c x)\right )^2+(2 b c) \int \frac{x \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{c}+x \left (a+b \cos ^{-1}(c x)\right )^2-\left (2 b^2\right ) \int 1 \, dx\\ &=-2 b^2 x-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{c}+x \left (a+b \cos ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.0615706, size = 76, normalized size = 1.62 \[ x \left (a^2-2 b^2\right )-\frac{2 a b \sqrt{1-c^2 x^2}}{c}+\frac{2 b \cos ^{-1}(c x) \left (a c x-b \sqrt{1-c^2 x^2}\right )}{c}+b^2 x \cos ^{-1}(c x)^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 74, normalized size = 1.6 \begin{align*}{\frac{1}{c} \left ( cx{a}^{2}+{b}^{2} \left ( cx \left ( \arccos \left ( cx \right ) \right ) ^{2}-2\,cx-2\,\arccos \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,ab \left ( cx\arccos \left ( cx \right ) -\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43825, size = 99, normalized size = 2.11 \begin{align*} b^{2} x \arccos \left (c x\right )^{2} - 2 \, b^{2}{\left (x + \frac{\sqrt{-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{2} x + \frac{2 \,{\left (c x \arccos \left (c x\right ) - \sqrt{-c^{2} x^{2} + 1}\right )} a b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41873, size = 159, normalized size = 3.38 \begin{align*} \frac{b^{2} c x \arccos \left (c x\right )^{2} + 2 \, a b c x \arccos \left (c x\right ) +{\left (a^{2} - 2 \, b^{2}\right )} c x - 2 \, \sqrt{-c^{2} x^{2} + 1}{\left (b^{2} \arccos \left (c x\right ) + a b\right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.331679, size = 87, normalized size = 1.85 \begin{align*} \begin{cases} a^{2} x + 2 a b x \operatorname{acos}{\left (c x \right )} - \frac{2 a b \sqrt{- c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname{acos}^{2}{\left (c x \right )} - 2 b^{2} x - \frac{2 b^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{acos}{\left (c x \right )}}{c} & \text{for}\: c \neq 0 \\x \left (a + \frac{\pi b}{2}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14457, size = 101, normalized size = 2.15 \begin{align*} b^{2} x \arccos \left (c x\right )^{2} + 2 \, a b x \arccos \left (c x\right ) + a^{2} x - 2 \, b^{2} x - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b^{2} \arccos \left (c x\right )}{c} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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