Integrand size = 12, antiderivative size = 76 \[ \int x (a+b \arcsin (c x))^2 \, dx=-\frac {1}{4} b^2 x^2+\frac {b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {(a+b \arcsin (c x))^2}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^2 \]
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Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4723, 4795, 4737, 30} \[ \int x (a+b \arcsin (c x))^2 \, dx=\frac {b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {(a+b \arcsin (c x))^2}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^2-\frac {1}{4} b^2 x^2 \]
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Rule 30
Rule 4723
Rule 4737
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \arcsin (c x))^2-(b c) \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {1}{2} x^2 (a+b \arcsin (c x))^2-\frac {1}{2} b^2 \int x \, dx-\frac {b \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c} \\ & = -\frac {1}{4} b^2 x^2+\frac {b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {(a+b \arcsin (c x))^2}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^2 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int x (a+b \arcsin (c x))^2 \, dx=-\frac {b^2 c^2 x^2-2 b c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+(a+b \arcsin (c x))^2-2 c^2 x^2 (a+b \arcsin (c x))^2}{4 c^2} \]
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Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.57
method | result | size |
parts | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{2}+\frac {\arcsin \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2}-\frac {\arcsin \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}\right )}{c^{2}}+\frac {2 a b \left (\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(119\) |
derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{2}+\frac {\arcsin \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2}-\frac {\arcsin \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}\right )+2 a b \left (\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(120\) |
default | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{2}+\frac {\arcsin \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2}-\frac {\arcsin \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}\right )+2 a b \left (\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(120\) |
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Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30 \[ \int x (a+b \arcsin (c x))^2 \, dx=\frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} x^{2} + {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (2 \, a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right ) + 2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.66 \[ \int x (a+b \arcsin (c x))^2 \, dx=\begin {cases} \frac {a^{2} x^{2}}{2} + a b x^{2} \operatorname {asin}{\left (c x \right )} + \frac {a b x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} x^{2}}{4} + \frac {b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{2}}{2} & \text {otherwise} \end {cases} \]
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\[ \int x (a+b \arcsin (c x))^2 \, dx=\int { {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (66) = 132\).
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.04 \[ \int x (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {-c^{2} x^{2} + 1} b^{2} x \arcsin \left (c x\right )}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b x}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a b \arcsin \left (c x\right )}{c^{2}} + \frac {b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2}}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2}}{4 \, c^{2}} + \frac {a b \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {b^{2}}{8 \, c^{2}} \]
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Timed out. \[ \int x (a+b \arcsin (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2 \,d x \]
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