Integrand size = 8, antiderivative size = 22 \[ \int e^x \arcsin \left (e^x\right ) \, dx=\sqrt {1-e^{2 x}}+e^x \arcsin \left (e^x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2225, 4928, 2278, 32} \[ \int e^x \arcsin \left (e^x\right ) \, dx=e^x \arcsin \left (e^x\right )+\sqrt {1-e^{2 x}} \]
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Rule 32
Rule 2225
Rule 2278
Rule 4928
Rubi steps \begin{align*} \text {integral}& = e^x \arcsin \left (e^x\right )-\int \frac {e^{2 x}}{\sqrt {1-e^{2 x}}} \, dx \\ & = e^x \arcsin \left (e^x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x}} \, dx,x,e^{2 x}\right ) \\ & = \sqrt {1-e^{2 x}}+e^x \arcsin \left (e^x\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int e^x \arcsin \left (e^x\right ) \, dx=\sqrt {1-e^{2 x}}+e^x \arcsin \left (e^x\right ) \]
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Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \({\mathrm e}^{x} \arcsin \left ({\mathrm e}^{x}\right )+\sqrt {1-{\mathrm e}^{2 x}}\) | \(18\) |
default | \({\mathrm e}^{x} \arcsin \left ({\mathrm e}^{x}\right )+\sqrt {1-{\mathrm e}^{2 x}}\) | \(18\) |
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int e^x \arcsin \left (e^x\right ) \, dx=\arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \]
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int e^x \arcsin \left (e^x\right ) \, dx=\sqrt {1 - e^{2 x}} + e^{x} \operatorname {asin}{\left (e^{x} \right )} \]
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Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int e^x \arcsin \left (e^x\right ) \, dx=\arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \]
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Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int e^x \arcsin \left (e^x\right ) \, dx=\arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \]
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Time = 0.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int e^x \arcsin \left (e^x\right ) \, dx=\sqrt {1-{\mathrm {e}}^{2\,x}}+\mathrm {asin}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^x \]
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