Optimal. Leaf size=35 \[ \frac {1}{4} e^{x^2} \sqrt {1-e^{2 x^2}}+\frac {1}{4} \sin ^{-1}\left (e^{x^2}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6715, 2249, 195, 216} \[ \frac {1}{4} e^{x^2} \sqrt {1-e^{2 x^2}}+\frac {1}{4} \sin ^{-1}\left (e^{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 2249
Rule 6715
Rubi steps
\begin {align*} \int e^{x^2} \sqrt {1-e^{2 x^2}} x \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int e^x \sqrt {1-e^{2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {1-x^2} \, dx,x,e^{x^2}\right )\\ &=\frac {1}{4} e^{x^2} \sqrt {1-e^{2 x^2}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,e^{x^2}\right )\\ &=\frac {1}{4} e^{x^2} \sqrt {1-e^{2 x^2}}+\frac {1}{4} \sin ^{-1}\left (e^{x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.91 \[ \frac {1}{4} \left (e^{x^2} \sqrt {1-e^{2 x^2}}+\sin ^{-1}\left (e^{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 43, normalized size = 1.23 \[ \frac {1}{4} \, \sqrt {-e^{\left (2 \, x^{2}\right )} + 1} e^{\left (x^{2}\right )} - \frac {1}{2} \, \arctan \left ({\left (\sqrt {-e^{\left (2 \, x^{2}\right )} + 1} - 1\right )} e^{\left (-x^{2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 26, normalized size = 0.74 \[ \frac {1}{4} \, \sqrt {-e^{\left (2 \, x^{2}\right )} + 1} e^{\left (x^{2}\right )} + \frac {1}{4} \, \arcsin \left (e^{\left (x^{2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 27, normalized size = 0.77 \[ \frac {\arcsin \left ({\mathrm e}^{x^{2}}\right )}{4}+\frac {\sqrt {-{\mathrm e}^{2 x^{2}}+1}\, {\mathrm e}^{x^{2}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.20, size = 26, normalized size = 0.74 \[ \frac {1}{4} \, \sqrt {-e^{\left (2 \, x^{2}\right )} + 1} e^{\left (x^{2}\right )} + \frac {1}{4} \, \arcsin \left (e^{\left (x^{2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 26, normalized size = 0.74 \[ \frac {\mathrm {asin}\left ({\mathrm {e}}^{x^2}\right )}{4}+\frac {{\mathrm {e}}^{x^2}\,\sqrt {1-{\mathrm {e}}^{2\,x^2}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 59.22, size = 0, normalized size = 0.00 \[ \text {NaN} \]
Verification of antiderivative is not currently implemented for this CAS.
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