Optimal. Leaf size=82 \[ -6 a b^2 x+\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c}+x \left (a+b \sin ^{-1}(c x)\right )^3-\frac{6 b^3 \sqrt{1-c^2 x^2}}{c}-6 b^3 x \sin ^{-1}(c x) \]
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Rubi [A] time = 0.108703, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4619, 4677, 261} \[ -6 a b^2 x+\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c}+x \left (a+b \sin ^{-1}(c x)\right )^3-\frac{6 b^3 \sqrt{1-c^2 x^2}}{c}-6 b^3 x \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 4619
Rule 4677
Rule 261
Rubi steps
\begin{align*} \int \left (a+b \sin ^{-1}(c x)\right )^3 \, dx &=x \left (a+b \sin ^{-1}(c x)\right )^3-(3 b c) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c}+x \left (a+b \sin ^{-1}(c x)\right )^3-\left (6 b^2\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-6 a b^2 x+\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c}+x \left (a+b \sin ^{-1}(c x)\right )^3-\left (6 b^3\right ) \int \sin ^{-1}(c x) \, dx\\ &=-6 a b^2 x-6 b^3 x \sin ^{-1}(c x)+\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c}+x \left (a+b \sin ^{-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 \sin ^{-1}(c x)+\frac{3 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c}+x \left (a+b \sin ^{-1}(c x)\right )^3\\ \end{align*}
Mathematica [A] time = 0.0881327, size = 77, normalized size = 0.94 \[ \frac{3 b \left (\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b \left (a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )\right )}{c}+x \left (a+b \sin ^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 132, normalized size = 1.6 \begin{align*}{\frac{1}{c} \left ( cx{a}^{3}+{b}^{3} \left ( cx \left ( \arcsin \left ( cx \right ) \right ) ^{3}+3\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}\sqrt{-{c}^{2}{x}^{2}+1}-6\,\sqrt{-{c}^{2}{x}^{2}+1}-6\,cx\arcsin \left ( cx \right ) \right ) +3\,a{b}^{2} \left ( cx \left ( \arcsin \left ( cx \right ) \right ) ^{2}-2\,cx+2\,\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,{a}^{2}b \left ( cx\arcsin \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 = 1.86907, size = 190, normalized size = 2.32 \begin{align*} b^{3} x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} x \arcsin \left (c x\right )^{2} + 3 \,{\left (\frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )^{2}}{c} - \frac{2 \,{\left (c x \arcsin \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} \arcsin \left (c x\right )}{c}\right )} + a^{3} x + \frac{3 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a^{2} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79325, size = 262, normalized size = 3.2 \begin{align*} \frac{b^{3} c x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} c x \arcsin \left (c x\right )^{2} + 3 \,{\left (a^{2} b - 2 \, b^{3}\right )} c x \arcsin \left (c x\right ) +{\left (a^{3} - 6 \, a b^{2}\right )} c x + 3 \,{\left (b^{3} \arcsin \left (c x\right )^{2} + 2 \, a b^{2} \arcsin \left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt{-c^{2} x^{2} + 1}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.2092, size = 160, normalized size = 1.95 \begin{align*} \begin{cases} a^{3} x + 3 a^{2} b x \operatorname{asin}{\left (c x \right )} + \frac{3 a^{2} b \sqrt{- c^{2} x^{2} + 1}}{c} + 3 a b^{2} x \operatorname{asin}^{2}{\left (c x \right )} - 6 a b^{2} x + \frac{6 a b^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{c} + b^{3} x \operatorname{asin}^{3}{\left (c x \right )} - 6 b^{3} x \operatorname{asin}{\left (c x \right )} + \frac{3 b^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (c x \right )}}{c} - \frac{6 b^{3} \sqrt{- c^{2} x^{2} + 1}}{c} & \text{for}\: c \neq 0 \\a^{3} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37369, size = 203, normalized size = 2.48 \begin{align*} b^{3} x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} x \arcsin \left (c x\right )^{2} + 3 \, a^{2} b x \arcsin \left (c x\right ) - 6 \, b^{3} x \arcsin \left (c x\right ) + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} b^{3} \arcsin \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} \arcsin \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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