Optimal. Leaf size=82 \[ -6 a b^2 x-\frac {3 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3+\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^3 x \cos ^{-1}(c x) \]
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Rubi [A] time = 0.11, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4620, 4678, 261} \[ -6 a b^2 x-\frac {3 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3+\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^3 x \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 261
Rule 4620
Rule 4678
Rubi steps
\begin {align*} \int \left (a+b \cos ^{-1}(c x)\right )^3 \, dx &=x \left (a+b \cos ^{-1}(c x)\right )^3+(3 b c) \int \frac {x \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {3 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3-\left (6 b^2\right ) \int \left (a+b \cos ^{-1}(c x)\right ) \, dx\\ &=-6 a b^2 x-\frac {3 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3-\left (6 b^3\right ) \int \cos ^{-1}(c x) \, dx\\ &=-6 a b^2 x-6 b^3 x \cos ^{-1}(c x)-\frac {3 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3-\left (6 b^3 c\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx\\ &=-6 a b^2 x+\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^3 x \cos ^{-1}(c x)-\frac {3 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{c}+x \left (a+b \cos ^{-1}(c x)\right )^3\\ \end {align*}
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Mathematica [A] time = 0.12, size = 128, normalized size = 1.56 \[ \frac {-3 b \left (a^2-2 b^2\right ) \sqrt {1-c^2 x^2}+3 b \cos ^{-1}(c x) \left (a^2 c x-2 a b \sqrt {1-c^2 x^2}-2 b^2 c x\right )+a c x \left (a^2-6 b^2\right )+3 b^2 \cos ^{-1}(c x)^2 \left (a c x-b \sqrt {1-c^2 x^2}\right )+b^3 c x \cos ^{-1}(c x)^3}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 108, normalized size = 1.32 \[ \frac {b^{3} c x \arccos \left (c x\right )^{3} + 3 \, a b^{2} c x \arccos \left (c x\right )^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} c x \arccos \left (c x\right ) + {\left (a^{3} - 6 \, a b^{2}\right )} c x - 3 \, {\left (b^{3} \arccos \left (c x\right )^{2} + 2 \, a b^{2} \arccos \left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 150, normalized size = 1.83 \[ b^{3} x \arccos \left (c x\right )^{3} + 3 \, a b^{2} x \arccos \left (c x\right )^{2} + 3 \, a^{2} b x \arccos \left (c x\right ) - 6 \, b^{3} x \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{c} + a^{3} x - 6 \, a b^{2} x - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{c} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{c} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 134, normalized size = 1.63 \[ \frac {c x \,a^{3}+b^{3} \left (c x \arccos \left (c x \right )^{3}-3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )+3 a \,b^{2} \left (c x \arccos \left (c x \right )^{2}-2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 144, normalized size = 1.76 \[ b^{3} x \arccos \left (c x\right )^{3} + 3 \, a b^{2} x \arccos \left (c x\right )^{2} - 3 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )^{2}}{c} + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )}}{c}\right )} b^{3} - 6 \, a b^{2} {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{3} x + \frac {3 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a^{2} b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 164, normalized size = 2.00 \[ \left \{\begin {array}{cl} x\,\left (a^3+\frac {3\,\pi \,a^2\,b}{2}+\frac {3\,\pi ^2\,a\,b^2}{4}+\frac {\pi ^3\,b^3}{8}\right ) & \text {\ if\ \ }c=0\\ a^3\,x-b^3\,x\,\left (6\,\mathrm {acos}\left (c\,x\right )-{\mathrm {acos}\left (c\,x\right )}^3\right )-\frac {3\,a^2\,b\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c}+3\,a\,b^2\,x\,\left ({\mathrm {acos}\left (c\,x\right )}^2-2\right )-\frac {b^3\,\sqrt {1-c^2\,x^2}\,\left (3\,{\mathrm {acos}\left (c\,x\right )}^2-6\right )}{c}-\frac {6\,a\,b^2\,\mathrm {acos}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 165, normalized size = 2.01 \[ \begin {cases} a^{3} x + 3 a^{2} b x \operatorname {acos}{\left (c x \right )} - \frac {3 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{c} + 3 a b^{2} x \operatorname {acos}^{2}{\left (c x \right )} - 6 a b^{2} x - \frac {6 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{c} + b^{3} x \operatorname {acos}^{3}{\left (c x \right )} - 6 b^{3} x \operatorname {acos}{\left (c x \right )} - \frac {3 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{c} + \frac {6 b^{3} \sqrt {- c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\x \left (a + \frac {\pi b}{2}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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