Optimal. Leaf size=4 \[ \tan ^{-1}\left (e^x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2249, 203} \[ \tan ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{1+e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (e^x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 4, normalized size = 1.00 \[ \tan ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 3, normalized size = 0.75 \[ \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 3, normalized size = 0.75 \[ \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 4, normalized size = 1.00 \[ \arctan \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 3, normalized size = 0.75 \[ \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.47, size = 3, normalized size = 0.75 \[ \mathrm {atan}\left ({\mathrm {e}}^x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.11, size = 15, normalized size = 3.75 \[ \operatorname {RootSum} {\left (4 z^{2} + 1, \left (i \mapsto i \log {\left (2 i + e^{x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________