Integrand size = 18, antiderivative size = 158 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {40 c \sqrt {1-a^2 x^2}}{9 a}-\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac {14}{3} c x \arcsin (a x)+\frac {2}{9} a^2 c x^3 \arcsin (a x)+\frac {2 c \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2}{3} c x \arcsin (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3 \]
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Time = 0.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4743, 4715, 4767, 267, 4739, 455, 45} \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=\frac {2}{9} a^2 c x^3 \arcsin (a x)+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2 c \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}-\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac {40 c \sqrt {1-a^2 x^2}}{9 a}+\frac {2}{3} c x \arcsin (a x)^3-\frac {14}{3} c x \arcsin (a x) \]
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Rule 45
Rule 267
Rule 455
Rule 4715
Rule 4739
Rule 4743
Rule 4767
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {1}{3} (2 c) \int \arcsin (a x)^3 \, dx-(a c) \int x \sqrt {1-a^2 x^2} \arcsin (a x)^2 \, dx \\ & = \frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2}{3} c x \arcsin (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3-\frac {1}{3} (2 c) \int \left (1-a^2 x^2\right ) \arcsin (a x) \, dx-(2 a c) \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2}{3} c x \arcsin (a x)+\frac {2}{9} a^2 c x^3 \arcsin (a x)+\frac {2 c \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2}{3} c x \arcsin (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3-(4 c) \int \arcsin (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 \arcsin (a x)+\frac {2}{9} a^2 c x^3 \arcsin (a x)+\frac {2 c \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2}{3} c x \arcsin (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {1}{3} (a c) \text {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 \arcsin (a x)+\frac {2}{9} a^2 c x^3 \arcsin (a x)+\frac {2 c \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2}{3} c x \arcsin (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {1}{3} (a c) \text {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 \arcsin (a x)+\frac {2}{9} a^2 c x^3 \arcsin (a x)+\frac {2 c \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2}{3} c x \arcsin (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.64 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=\frac {c \left (2 \sqrt {1-a^2 x^2} \left (-61+a^2 x^2\right )+6 a x \left (-21+a^2 x^2\right ) \arcsin (a x)-9 \sqrt {1-a^2 x^2} \left (-7+a^2 x^2\right ) \arcsin (a x)^2-9 a x \left (-3+a^2 x^2\right ) \arcsin (a x)^3\right )}{27 a} \]
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Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {c \left (9 a^{3} x^{3} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-27 a x \arcsin \left (a x \right )^{3}-6 a^{3} x^{3} \arcsin \left (a x \right )-63 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arcsin \left (a x \right )+122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\) | \(132\) |
default | \(-\frac {c \left (9 a^{3} x^{3} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-27 a x \arcsin \left (a x \right )^{3}-6 a^{3} x^{3} \arcsin \left (a x \right )-63 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arcsin \left (a x \right )+122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\) | \(132\) |
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Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.60 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \arcsin \left (a x\right )^{3} - 6 \, {\left (a^{3} c x^{3} - 21 \, a c x\right )} \arcsin \left (a x\right ) - {\left (2 \, a^{2} c x^{2} - 9 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \arcsin \left (a x\right )^{2} - 122 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a} \]
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Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.95 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=\begin {cases} - \frac {a^{2} c x^{3} \operatorname {asin}^{3}{\left (a x \right )}}{3} + \frac {2 a^{2} c x^{3} \operatorname {asin}{\left (a x \right )}}{9} - \frac {a c x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3} + \frac {2 a c x^{2} \sqrt {- a^{2} x^{2} + 1}}{27} + c x \operatorname {asin}^{3}{\left (a x \right )} - \frac {14 c x \operatorname {asin}{\left (a x \right )}}{3} + \frac {7 c \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3 a} - \frac {122 c \sqrt {- a^{2} x^{2} + 1}}{27 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {1}{3} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} - \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \arcsin \left (a x\right )^{3} + \frac {2}{27} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} + \frac {3 \, {\left (a^{2} c x^{3} - 21 \, c x\right )} \arcsin \left (a x\right )}{a} - \frac {61 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \]
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Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.88 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {1}{3} \, {\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right )^{3} + \frac {2}{3} \, c x \arcsin \left (a x\right )^{3} + \frac {2}{9} \, {\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right ) + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c \arcsin \left (a x\right )^{2}}{3 \, a} - \frac {40}{9} \, c x \arcsin \left (a x\right ) + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c \arcsin \left (a x\right )^{2}}{a} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{27 \, a} - \frac {40 \, \sqrt {-a^{2} x^{2} + 1} c}{9 \, a} \]
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Timed out. \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=\int {\mathrm {asin}\left (a\,x\right )}^3\,\left (c-a^2\,c\,x^2\right ) \,d x \]
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