Integrand size = 21, antiderivative size = 10 \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\frac {e^{\arcsin (a x)}}{a} \]
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Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4920, 6820, 6852, 2225} \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\frac {e^{\arcsin (a x)}}{a} \]
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Rule 2225
Rule 4920
Rule 6820
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^x \cos (x)}{\sqrt {1-\sin ^2(x)}} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int e^x \sqrt {\cos ^2(x)} \sec (x) \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int e^x \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {e^{\arcsin (a x)}}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\frac {e^{\arcsin (a x)}}{a} \]
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Time = 0.35 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {{\mathrm e}^{\arcsin \left (a x \right )}}{a}\) | \(10\) |
default | \(\frac {{\mathrm e}^{\arcsin \left (a x \right )}}{a}\) | \(10\) |
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Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \]
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Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {e^{\operatorname {asin}{\left (a x \right )}}}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \]
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Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \]
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Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx=\frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{a} \]
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