Optimal. Leaf size=20 \[ \frac {e^{2 x}}{2}-\frac {1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2248, 321, 203} \[ \frac {e^{2 x}}{2}-\frac {1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{6 x}}{1+e^{4 x}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,e^{2 x}\right )\\ &=\frac {e^{2 x}}{2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^{2 x}\right )\\ &=\frac {e^{2 x}}{2}-\frac {1}{2} \tan ^{-1}\left (e^{2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.90 \[ \frac {1}{2} \left (e^{2 x}-\tan ^{-1}\left (e^{2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 14, normalized size = 0.70 \[ -\frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 14, normalized size = 0.70 \[ -\frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 15, normalized size = 0.75 \[ -\frac {\arctan \left ({\mathrm e}^{2 x}\right )}{2}+\frac {{\mathrm e}^{2 x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 14, normalized size = 0.70 \[ -\frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 14, normalized size = 0.70 \[ \frac {{\mathrm {e}}^{2\,x}}{2}-\frac {\mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 24, normalized size = 1.20 \[ \frac {e^{2 x}}{2} + \operatorname {RootSum} {\left (16 z^{2} + 1, \left (i \mapsto i \log {\left (- 4 i + e^{2 x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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