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 = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 8
Rule 4620
Rule 4678
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.43, size = 65, normalized size = 1.38 \[ \frac {b^{2} c x \arccos \left (c x\right )^{2} + 2 \, a b c x \arccos \left (c x\right ) + {\left (a^{2} - 2 \, b^{2}\right )} c x - 2 \, \sqrt {-c^{2} x^{2} + 1} {\left (b^{2} \arccos \left (c x\right ) + a b\right )}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 75, normalized size = 1.60 \[ b^{2} x \arccos \left (c x\right )^{2} + 2 \, a b x \arccos \left (c x\right ) + a^{2} x - 2 \, b^{2} x - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \arccos \left (c x\right )}{c} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 74, normalized size = 1.57 \[ \frac {c x \,a^{2}+b^{2} \left (c x \arccos \left (c x \right )^{2}-2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+2 a b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 73, normalized size = 1.55 \[ b^{2} x \arccos \left (c x\right )^{2} - 2 \, b^{2} {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{2} x + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 96, normalized size = 2.04 \[ \left \{\begin {array}{cl} x\,\left (a^2+\pi \,a\,b+\frac {\pi ^2\,b^2}{4}\right ) & \text {\ if\ \ }c=0\\ a^2\,x+b^2\,x\,\left ({\mathrm {acos}\left (c\,x\right )}^2-2\right )-\frac {2\,b^2\,\mathrm {acos}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c}-\frac {2\,a\,b\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 87, normalized size = 1.85 \[ \begin {cases} a^{2} x + 2 a b x \operatorname {acos}{\left (c x \right )} - \frac {2 a b \sqrt {- c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname {acos}^{2}{\left (c x \right )} - 2 b^{2} x - \frac {2 b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\x \left (a + \frac {\pi b}{2}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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