Optimal. Leaf size=158 \[ -\frac{2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac{40 c \sqrt{1-a^2 x^2}}{9 a}+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sin ^{-1}(a x)^3-\frac{14}{3} c x \sin ^{-1}(a x) \]
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Rubi [A] time = 0.210908, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {4649, 4619, 4677, 261, 4645, 444, 43} \[ -\frac{2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac{40 c \sqrt{1-a^2 x^2}}{9 a}+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sin ^{-1}(a x)^3-\frac{14}{3} c x \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4649
Rule 4619
Rule 4677
Rule 261
Rule 4645
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^3 \, dx &=\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{1}{3} (2 c) \int \sin ^{-1}(a x)^3 \, dx-(a c) \int x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \, dx\\ &=\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3-\frac{1}{3} (2 c) \int \left (1-a^2 x^2\right ) \sin ^{-1}(a x) \, dx-(2 a c) \int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3-(4 c) \int \sin ^{-1}(a x) \, dx+\frac{1}{3} (2 a c) \int \frac{x \left (1-\frac{a^2 x^2}{3}\right )}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{14}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{1}{3} (a c) \operatorname{Subst}\left (\int \frac{1-\frac{a^2 x}{3}}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )+(4 a c) \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 c \sqrt{1-a^2 x^2}}{a}-\frac{14}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{1}{3} (a c) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-a^2 x}}+\frac{1}{3} \sqrt{1-a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{40 c \sqrt{1-a^2 x^2}}{9 a}-\frac{2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac{14}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3\\ \end{align*}
Mathematica [A] time = 0.0896877, size = 101, normalized size = 0.64 \[ \frac{c \left (2 \sqrt{1-a^2 x^2} \left (a^2 x^2-61\right )-9 a x \left (a^2 x^2-3\right ) \sin ^{-1}(a x)^3-9 \sqrt{1-a^2 x^2} \left (a^2 x^2-7\right ) \sin ^{-1}(a x)^2+6 a x \left (a^2 x^2-21\right ) \sin ^{-1}(a x)\right )}{27 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 132, normalized size = 0.8 \begin{align*} -{\frac{c}{27\,a} \left ( 9\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}+9\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}{a}^{2}{x}^{2}-27\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{3}-6\,{a}^{3}{x}^{3}\arcsin \left ( ax \right ) -63\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}+126\,ax\arcsin \left ( ax \right ) +122\,\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58552, size = 173, normalized size = 1.09 \begin{align*} -\frac{1}{3} \,{\left (\sqrt{-a^{2} x^{2} + 1} c x^{2} - \frac{7 \, \sqrt{-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} - \frac{1}{3} \,{\left (a^{2} c x^{3} - 3 \, c x\right )} \arcsin \left (a x\right )^{3} + \frac{2}{27} \,{\left (\sqrt{-a^{2} x^{2} + 1} c x^{2} + \frac{3 \,{\left (a^{2} c x^{3} - 21 \, c x\right )} \arcsin \left (a x\right )}{a} - \frac{61 \, \sqrt{-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61613, size = 225, normalized size = 1.42 \begin{align*} -\frac{9 \,{\left (a^{3} c x^{3} - 3 \, a c x\right )} \arcsin \left (a x\right )^{3} - 6 \,{\left (a^{3} c x^{3} - 21 \, a c x\right )} \arcsin \left (a x\right ) -{\left (2 \, a^{2} c x^{2} - 9 \,{\left (a^{2} c x^{2} - 7 \, c\right )} \arcsin \left (a x\right )^{2} - 122 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{27 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.45054, size = 150, normalized size = 0.95 \begin{align*} \begin{cases} - \frac{a^{2} c x^{3} \operatorname{asin}^{3}{\left (a x \right )}}{3} + \frac{2 a^{2} c x^{3} \operatorname{asin}{\left (a x \right )}}{9} - \frac{a c x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3} + \frac{2 a c x^{2} \sqrt{- a^{2} x^{2} + 1}}{27} + c x \operatorname{asin}^{3}{\left (a x \right )} - \frac{14 c x \operatorname{asin}{\left (a x \right )}}{3} + \frac{7 c \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3 a} - \frac{122 c \sqrt{- a^{2} x^{2} + 1}}{27 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38332, size = 188, normalized size = 1.19 \begin{align*} -\frac{1}{3} \,{\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right )^{3} + \frac{2}{3} \, c x \arcsin \left (a x\right )^{3} + \frac{2}{9} \,{\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right ) + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c \arcsin \left (a x\right )^{2}}{3 \, a} - \frac{40}{9} \, c x \arcsin \left (a x\right ) + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} c \arcsin \left (a x\right )^{2}}{a} - \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c}{27 \, a} - \frac{40 \, \sqrt{-a^{2} x^{2} + 1} c}{9 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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