Optimal. Leaf size=178 \[ -\frac {4 a b^2 x}{3 c^2}-\frac {2}{9} b^2 x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac {b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c}-\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^3-\frac {4 b^3 x \cos ^{-1}(c x)}{3 c^2}-\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}+\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3} \]
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Rubi [A] time = 0.30, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 43
Rule 261
Rule 266
Rule 4620
Rule 4628
Rule 4678
Rule 4708
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*}
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Mathematica [A] time = 0.25, size = 218, normalized size = 1.22 \[ \frac {9 a^3 c^3 x^3-3 b \cos ^{-1}(c x) \left (-9 a^2 c^3 x^3+6 a b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )+2 b^2 c x \left (c^2 x^2+6\right )\right )-9 a^2 b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )-6 a b^2 c x \left (c^2 x^2+6\right )-9 b^2 \cos ^{-1}(c x)^2 \left (b \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right )-3 a c^3 x^3\right )+9 b^3 c^3 x^3 \cos ^{-1}(c x)^3+2 b^3 \sqrt {1-c^2 x^2} \left (c^2 x^2+20\right )}{27 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 195, normalized size = 1.10 \[ \frac {9 \, b^{3} c^{3} x^{3} \arccos \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arccos \left (c x\right )^{2} + 3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} x^{3} - 36 \, a b^{2} c x + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} x^{3} - 12 \, b^{3} c x\right )} \arccos \left (c x\right ) - {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} x^{2} + 18 \, a^{2} b - 40 \, b^{3} + 9 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.98, size = 289, normalized size = 1.62 \[ \frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + a^{2} b x^{3} \arccos \left (c x\right ) - \frac {2}{9} \, b^{3} x^{3} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b^{3} x^{2} \arccos \left (c x\right )^{2}}{3 \, c} + \frac {1}{3} \, a^{3} x^{3} - \frac {2}{9} \, a b^{2} x^{3} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x^{2} \arccos \left (c x\right )}{3 \, c} - \frac {\sqrt {-c^{2} x^{2} + 1} a^{2} b x^{2}}{3 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x^{2}}{27 \, c} - \frac {4 \, b^{3} x \arccos \left (c x\right )}{3 \, c^{2}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{3 \, c^{3}} - \frac {4 \, a b^{2} x}{3 \, c^{2}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{3 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{3 \, c^{3}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{27 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 235, normalized size = 1.32 \[ \frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {\arccos \left (c x \right )^{3} c^{3} x^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 273, normalized size = 1.53 \[ \frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} - \frac {1}{27} \, {\left (9 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right )^{2} - 2 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} - \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arccos \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.32, size = 333, normalized size = 1.87 \[ \begin {cases} \frac {a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname {acos}{\left (c x \right )} - \frac {a^{2} b x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} - \frac {2 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + a b^{2} x^{3} \operatorname {acos}^{2}{\left (c x \right )} - \frac {2 a b^{2} x^{3}}{9} - \frac {2 a b^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3 c} - \frac {4 a b^{2} x}{3 c^{2}} - \frac {4 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3 c^{3}} + \frac {b^{3} x^{3} \operatorname {acos}^{3}{\left (c x \right )}}{3} - \frac {2 b^{3} x^{3} \operatorname {acos}{\left (c x \right )}}{9} - \frac {b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{3 c} + \frac {2 b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{27 c} - \frac {4 b^{3} x \operatorname {acos}{\left (c x \right )}}{3 c^{2}} - \frac {2 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{3 c^{3}} + \frac {40 b^{3} \sqrt {- c^{2} x^{2} + 1}}{27 c^{3}} & \text {for}\: c \neq 0 \\\frac {x^{3} \left (a + \frac {\pi b}{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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