Optimal. Leaf size=18 \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {e^{2 x^2}-1}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6715, 2282, 63, 203} \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {e^{2 x^2}-1}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 2282
Rule 6715
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-1+e^{2 x^2}}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+e^{2 x}}} \, dx,x,x^2\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,e^{2 x^2}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+e^{2 x^2}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+e^{2 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {e^{2 x^2}-1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 13, normalized size = 0.72 \[ \frac {1}{2} \, \arctan \left (\sqrt {e^{\left (2 \, x^{2}\right )} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 13, normalized size = 0.72 \[ \frac {1}{2} \, \arctan \left (\sqrt {e^{\left (2 \, x^{2}\right )} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 0.78 \[ \frac {\arctan \left (\sqrt {{\mathrm e}^{2 x^{2}}-1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.24, size = 13, normalized size = 0.72 \[ \frac {1}{2} \, \arctan \left (\sqrt {e^{\left (2 \, x^{2}\right )} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.71, size = 13, normalized size = 0.72 \[ \frac {\mathrm {atan}\left (\sqrt {{\mathrm {e}}^{2\,x^2}-1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {\left (e^{x^{2}} - 1\right ) \left (e^{x^{2}} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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