Optimal. Leaf size=12 \[ \frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2249, 206} \[ \frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{16-e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,e^x\right )\\ &=\frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 12, normalized size = 1.00 \[ \frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 15, normalized size = 1.25 \[ \frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left (e^{x} - 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 16, normalized size = 1.33 \[ \frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left ({\left | e^{x} - 4 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 16, normalized size = 1.33 \[ -\frac {\ln \left ({\mathrm e}^{x}-4\right )}{8}+\frac {\ln \left ({\mathrm e}^{x}+4\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.05, size = 15, normalized size = 1.25 \[ \frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left (e^{x} - 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.65, size = 15, normalized size = 1.25 \[ \frac {\ln \left ({\mathrm {e}}^x+4\right )}{8}-\frac {\ln \left ({\mathrm {e}}^x-4\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.11, size = 15, normalized size = 1.25 \[ - \frac {\log {\left (e^{x} - 4 \right )}}{8} + \frac {\log {\left (e^{x} + 4 \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________