Optimal. Leaf size=178 \[ -\frac{4 a b^2 x}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac{2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}+\frac{14 b^3 \sqrt{1-c^2 x^2}}{9 c^3}-\frac{4 b^3 x \cos ^{-1}(c x)}{3 c^2} \]
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Rubi [A] time = 0.295393, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4628, 4708, 4678, 4620, 261, 266, 43} \[ -\frac{4 a b^2 x}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac{2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}+\frac{14 b^3 \sqrt{1-c^2 x^2}}{9 c^3}-\frac{4 b^3 x \cos ^{-1}(c x)}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 4708
Rule 4678
Rule 4620
Rule 261
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (a+b \cos ^{-1}(c x)\right )^3 \, dx &=\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3+(b c) \int \frac{x^3 \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac{1}{3} \left (2 b^2\right ) \int x^2 \left (a+b \cos ^{-1}(c x)\right ) \, dx+\frac{(2 b) \int \frac{x \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{3 c}\\ &=-\frac{2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac{\left (4 b^2\right ) \int \left (a+b \cos ^{-1}(c x)\right ) \, dx}{3 c^2}-\frac{1}{9} \left (2 b^3 c\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{4 a b^2 x}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac{\left (4 b^3\right ) \int \cos ^{-1}(c x) \, dx}{3 c^2}-\frac{1}{9} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{4 a b^2 x}{3 c^2}-\frac{4 b^3 x \cos ^{-1}(c x)}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac{\left (4 b^3\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{3 c}-\frac{1}{9} \left (b^3 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \sqrt{1-c^2 x}}-\frac{\sqrt{1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{4 a b^2 x}{3 c^2}+\frac{14 b^3 \sqrt{1-c^2 x^2}}{9 c^3}-\frac{2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac{4 b^3 x \cos ^{-1}(c x)}{3 c^2}-\frac{2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3\\ \end{align*}
Mathematica [A] time = 0.198356, size = 218, normalized size = 1.22 \[ \frac{-3 b \cos ^{-1}(c x) \left (-9 a^2 c^3 x^3+6 a b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )+2 b^2 c x \left (c^2 x^2+6\right )\right )-9 a^2 b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )+9 a^3 c^3 x^3-6 a b^2 c x \left (c^2 x^2+6\right )-9 b^2 \cos ^{-1}(c x)^2 \left (b \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right )-3 a c^3 x^3\right )+2 b^3 \sqrt{1-c^2 x^2} \left (c^2 x^2+20\right )+9 b^3 c^3 x^3 \cos ^{-1}(c x)^3}{27 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 235, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{{c}^{3}{x}^{3}{a}^{3}}{3}}+{b}^{3} \left ({\frac{{c}^{3}{x}^{3} \left ( \arccos \left ( cx \right ) \right ) ^{3}}{3}}-{\frac{ \left ( \arccos \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}+2 \right ) }{3}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{4}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{4\,cx\arccos \left ( cx \right ) }{3}}-{\frac{2\,{c}^{3}{x}^{3}\arccos \left ( cx \right ) }{9}}+{\frac{2\,{c}^{2}{x}^{2}+4}{27}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) +3\,a{b}^{2} \left ( 1/3\, \left ( \arccos \left ( cx \right ) \right ) ^{2}{c}^{3}{x}^{3}-2/9\,\arccos \left ( cx \right ) \left ({c}^{2}{x}^{2}+2 \right ) \sqrt{-{c}^{2}{x}^{2}+1}-{\frac{2\,{c}^{3}{x}^{3}}{27}}-4/9\,cx \right ) +3\,{a}^{2}b \left ( 1/3\,{c}^{3}{x}^{3}\arccos \left ( cx \right ) -1/9\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}-2/9\,\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 = 1.4924, size = 369, normalized size = 2.07 \begin{align*} \frac{1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + \frac{1}{3} \, a^{3} x^{3} + \frac{1}{3} \,{\left (3 \, x^{3} \arccos \left (c x\right ) - c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b - \frac{2}{9} \,{\left (3 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} - \frac{1}{27} \,{\left (9 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right )^{2} - 2 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} - \frac{3 \,{\left (c^{2} x^{3} + 6 \, x\right )} \arccos \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42432, size = 441, normalized size = 2.48 \begin{align*} \frac{9 \, b^{3} c^{3} x^{3} \arccos \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arccos \left (c x\right )^{2} + 3 \,{\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} x^{3} - 36 \, a b^{2} c x + 3 \,{\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} x^{3} - 12 \, b^{3} c x\right )} \arccos \left (c x\right ) -{\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} x^{2} + 18 \, a^{2} b - 40 \, b^{3} + 9 \,{\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \arccos \left (c x\right )^{2} + 18 \,{\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arccos \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{27 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.29767, size = 333, normalized size = 1.87 \begin{align*} \begin{cases} \frac{a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname{acos}{\left (c x \right )} - \frac{a^{2} b x^{2} \sqrt{- c^{2} x^{2} + 1}}{3 c} - \frac{2 a^{2} b \sqrt{- c^{2} x^{2} + 1}}{3 c^{3}} + a b^{2} x^{3} \operatorname{acos}^{2}{\left (c x \right )} - \frac{2 a b^{2} x^{3}}{9} - \frac{2 a b^{2} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{acos}{\left (c x \right )}}{3 c} - \frac{4 a b^{2} x}{3 c^{2}} - \frac{4 a b^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{acos}{\left (c x \right )}}{3 c^{3}} + \frac{b^{3} x^{3} \operatorname{acos}^{3}{\left (c x \right )}}{3} - \frac{2 b^{3} x^{3} \operatorname{acos}{\left (c x \right )}}{9} - \frac{b^{3} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (c x \right )}}{3 c} + \frac{2 b^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{27 c} - \frac{4 b^{3} x \operatorname{acos}{\left (c x \right )}}{3 c^{2}} - \frac{2 b^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (c x \right )}}{3 c^{3}} + \frac{40 b^{3} \sqrt{- c^{2} x^{2} + 1}}{27 c^{3}} & \text{for}\: c \neq 0 \\\frac{x^{3} \left (a + \frac{\pi b}{2}\right )^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.213, size = 390, normalized size = 2.19 \begin{align*} \frac{1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + a^{2} b x^{3} \arccos \left (c x\right ) - \frac{2}{9} \, b^{3} x^{3} \arccos \left (c x\right ) - \frac{\sqrt{-c^{2} x^{2} + 1} b^{3} x^{2} \arccos \left (c x\right )^{2}}{3 \, c} + \frac{1}{3} \, a^{3} x^{3} - \frac{2}{9} \, a b^{2} x^{3} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a b^{2} x^{2} \arccos \left (c x\right )}{3 \, c} - \frac{\sqrt{-c^{2} x^{2} + 1} a^{2} b x^{2}}{3 \, c} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b^{3} x^{2}}{27 \, c} - \frac{4 \, b^{3} x \arccos \left (c x\right )}{3 \, c^{2}} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{3 \, c^{3}} - \frac{4 \, a b^{2} x}{3 \, c^{2}} - \frac{4 \, \sqrt{-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{3 \, c^{3}} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a^{2} b}{3 \, c^{3}} + \frac{40 \, \sqrt{-c^{2} x^{2} + 1} b^{3}}{27 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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