Integrand size = 26, antiderivative size = 49 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {4737} \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \]
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Rule 4737
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} \arcsin (c x) \left (3 a^2+3 a b \arcsin (c x)+b^2 \arcsin (c x)^2\right )}{3 c \sqrt {d-c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs. \(2(43)=86\).
Time = 0.15 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.92
method | result | size |
default | \(\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{c d \left (c^{2} x^{2}-1\right )}\) | \(143\) |
parts | \(\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{c d \left (c^{2} x^{2}-1\right )}\) | \(143\) |
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b^{2} \arcsin \left (c x\right )^{3}}{3 \, c \sqrt {d}} + \frac {a b \arcsin \left (c x\right )^{2}}{c \sqrt {d}} + \frac {a^{2} \arcsin \left (c x\right )}{c \sqrt {d}} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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