Integrand size = 10, antiderivative size = 47 \[ \int (a+b \arcsin (c x))^2 \, dx=-2 b^2 x+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+x (a+b \arcsin (c x))^2 \]
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Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4715, 4767, 8} \[ \int (a+b \arcsin (c x))^2 \, dx=\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+x (a+b \arcsin (c x))^2-2 b^2 x \]
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Rule 8
Rule 4715
Rule 4767
Rubi steps \begin{align*} \text {integral}& = x (a+b \arcsin (c x))^2-(2 b c) \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+x (a+b \arcsin (c x))^2-\left (2 b^2\right ) \int 1 \, dx \\ & = -2 b^2 x+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+x (a+b \arcsin (c x))^2 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int (a+b \arcsin (c x))^2 \, dx=-2 b^2 x+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+x (a+b \arcsin (c x))^2 \]
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Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.53
method | result | size |
derivativedivides | \(\frac {c x \,a^{2}+b^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+2 a b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(72\) |
default | \(\frac {c x \,a^{2}+b^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+2 a b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(72\) |
parts | \(a^{2} x +\frac {b^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )}{c}+\frac {2 a b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(73\) |
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Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.38 \[ \int (a+b \arcsin (c x))^2 \, dx=\frac {b^{2} c x \arcsin \left (c x\right )^{2} + 2 \, a b c x \arcsin \left (c x\right ) + {\left (a^{2} - 2 \, b^{2}\right )} c x + 2 \, \sqrt {-c^{2} x^{2} + 1} {\left (b^{2} \arcsin \left (c x\right ) + a b\right )}}{c} \]
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Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.74 \[ \int (a+b \arcsin (c x))^2 \, dx=\begin {cases} a^{2} x + 2 a b x \operatorname {asin}{\left (c x \right )} + \frac {2 a b \sqrt {- c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} x + \frac {2 b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\a^{2} x & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.53 \[ \int (a+b \arcsin (c x))^2 \, dx=b^{2} x \arcsin \left (c x\right )^{2} - 2 \, b^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b}{c} \]
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.60 \[ \int (a+b \arcsin (c x))^2 \, dx=b^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b x \arcsin \left (c x\right ) + a^{2} x - 2 \, b^{2} x + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \arcsin \left (c x\right )}{c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b}{c} \]
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Time = 0.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.02 \[ \int (a+b \arcsin (c x))^2 \, dx=\left \{\begin {array}{cl} b^2\,\left (x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {\frac {1}{c^2}-x^2}\right )+a^2\,x+\frac {2\,a\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }0<c\\ a^2\,x+b^2\,x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+\frac {2\,b^2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c}+\frac {2\,a\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }\neg 0<c \end {array}\right . \]
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