Integrand size = 17, antiderivative size = 29 \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {1}{2} e^x \sqrt {1-e^{2 x}}+\frac {\arcsin \left (e^x\right )}{2} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2281, 201, 222} \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {\arcsin \left (e^x\right )}{2}+\frac {1}{2} e^x \sqrt {1-e^{2 x}} \]
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Rule 201
Rule 222
Rule 2281
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,e^x\right ) \\ & = \frac {1}{2} e^x \sqrt {1-e^{2 x}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,e^x\right ) \\ & = \frac {1}{2} e^x \sqrt {1-e^{2 x}}+\frac {1}{2} \sin ^{-1}\left (e^x\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {1}{2} e^x \sqrt {1-e^{2 x}}-\arctan \left (\frac {\sqrt {1-e^{2 x}}}{1+e^x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {\arcsin \left ({\mathrm e}^{x}\right )}{2}+\frac {{\mathrm e}^{x} \sqrt {1-{\mathrm e}^{2 x}}}{2}\) | \(21\) |
risch | \(-\frac {{\mathrm e}^{x} \left (-1+{\mathrm e}^{2 x}\right )}{2 \sqrt {1-{\mathrm e}^{2 x}}}+\frac {\arcsin \left ({\mathrm e}^{x}\right )}{2}\) | \(27\) |
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none
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 1} e^{x} - \arctan \left ({\left (\sqrt {-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {\sqrt {1 - e^{2 x}} e^{x}}{2} + \frac {\operatorname {asin}{\left (e^{x} \right )}}{2} \]
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 1} e^{x} + \frac {1}{2} \, \arcsin \left (e^{x}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 1} e^{x} + \frac {1}{2} \, \arcsin \left (e^{x}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int e^x \sqrt {1-e^{2 x}} \, dx=\frac {\mathrm {asin}\left ({\mathrm {e}}^x\right )}{2}+\frac {{\mathrm {e}}^x\,\sqrt {1-{\mathrm {e}}^{2\,x}}}{2} \]
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