Optimal. Leaf size=10 \[ e^x-\tan ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 10, 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} \[ e^x-\tan ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{3 x}}{1+e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,e^x\right )\\ &=e^x-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=e^x-\tan ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ e^x-\tan ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 8, normalized size = 0.80 \[ -\arctan \left (e^{x}\right ) + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 8, normalized size = 0.80 \[ -\arctan \left (e^{x}\right ) + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 9, normalized size = 0.90 \[ -\arctan \left ({\mathrm e}^{x}\right )+{\mathrm e}^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.93, size = 8, normalized size = 0.80 \[ -\arctan \left (e^{x}\right ) + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 8, normalized size = 0.80 \[ {\mathrm {e}}^x-\mathrm {atan}\left ({\mathrm {e}}^x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 19, normalized size = 1.90 \[ e^{x} + \operatorname {RootSum} {\left (4 z^{2} + 1, \left (i \mapsto i \log {\left (- 2 i + e^{x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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