Integrand size = 6, antiderivative size = 60 \[ \int \arcsin (a x)^3 \, dx=-\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3 \]
[Out]
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4715, 4767, 267} \[ \int \arcsin (a x)^3 \, dx=\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}-\frac {6 \sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)^3-6 x \arcsin (a x) \]
[In]
[Out]
Rule 267
Rule 4715
Rule 4767
Rubi steps \begin{align*} \text {integral}& = x \arcsin (a x)^3-(3 a) \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3-6 \int \arcsin (a x) \, dx \\ & = -6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3+(6 a) \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \arcsin (a x)^3 \, dx=-\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3 \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {a x \arcsin \left (a x \right )^{3}+3 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-6 \sqrt {-a^{2} x^{2}+1}-6 a x \arcsin \left (a x \right )}{a}\) | \(57\) |
default | \(\frac {a x \arcsin \left (a x \right )^{3}+3 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-6 \sqrt {-a^{2} x^{2}+1}-6 a x \arcsin \left (a x \right )}{a}\) | \(57\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \arcsin (a x)^3 \, dx=\frac {a x \arcsin \left (a x\right )^{3} - 6 \, a x \arcsin \left (a x\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{2} - 2\right )}}{a} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int \arcsin (a x)^3 \, dx=\begin {cases} x \operatorname {asin}^{3}{\left (a x \right )} - 6 x \operatorname {asin}{\left (a x \right )} + \frac {3 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{a} - \frac {6 \sqrt {- a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95 \[ \int \arcsin (a x)^3 \, dx=x \arcsin \left (a x\right )^{3} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac {6 \, {\left (a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}\right )}}{a} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \arcsin (a x)^3 \, dx=x \arcsin \left (a x\right )^{3} - 6 \, x \arcsin \left (a x\right ) + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.67 \[ \int \arcsin (a x)^3 \, dx=\frac {3\,\sqrt {1-a^2\,x^2}\,\left ({\mathrm {asin}\left (a\,x\right )}^2-2\right )}{a}+x\,\mathrm {asin}\left (a\,x\right )\,\left ({\mathrm {asin}\left (a\,x\right )}^2-6\right ) \]
[In]
[Out]