Optimal. Leaf size=22 \[ \sqrt {1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2194, 4844, 2246, 32} \[ \sqrt {1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 32
Rule 2194
Rule 2246
Rule 4844
Rubi steps
\begin {align*} \int e^x \sin ^{-1}\left (e^x\right ) \, dx &=e^x \sin ^{-1}\left (e^x\right )-\int \frac {e^{2 x}}{\sqrt {1-e^{2 x}}} \, dx\\ &=e^x \sin ^{-1}\left (e^x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x}} \, dx,x,e^{2 x}\right )\\ &=\sqrt {1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \sqrt {1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 17, normalized size = 0.77 \[ \arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.99, size = 17, normalized size = 0.77 \[ \arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 18, normalized size = 0.82 \[ {\mathrm e}^{x} \arcsin \left ({\mathrm e}^{x}\right )+\sqrt {1-{\mathrm e}^{2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 17, normalized size = 0.77 \[ \arcsin \left (e^{x}\right ) e^{x} + \sqrt {-e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 17, normalized size = 0.77 \[ \sqrt {1-{\mathrm {e}}^{2\,x}}+\mathrm {asin}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 17, normalized size = 0.77 \[ \sqrt {1 - e^{2 x}} + e^{x} \operatorname {asin}{\left (e^{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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