Optimal. Leaf size=13 \[ -\frac {1}{\left (1+3^x\right ) \log (3)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 32}
\begin {gather*} -\frac {1}{\left (3^x+1\right ) \log (3)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{2+3^{-x}+3^x} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,3^x\right )}{\log (3)}\\ &=-\frac {1}{\left (1+3^x\right ) \log (3)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{\left (1+3^x\right ) \log (3)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 14, normalized size = 1.08
method | result | size |
derivativedivides | \(-\frac {1}{\left (1+3^{x}\right ) \ln \left (3\right )}\) | \(14\) |
default | \(-\frac {1}{\left (1+3^{x}\right ) \ln \left (3\right )}\) | \(14\) |
risch | \(-\frac {1}{\left (1+3^{x}\right ) \ln \left (3\right )}\) | \(14\) |
norman | \(\frac {{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right ) \left ({\mathrm e}^{x \ln \left (3\right )}+1\right )}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 14, normalized size = 1.08 \begin {gather*} \frac {1}{{\left (\frac {1}{3^{x}} + 1\right )} \log \left (3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{3^{x} \log \left (3\right ) + \log \left (3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.03, size = 12, normalized size = 0.92 \begin {gather*} - \frac {1}{3^{x} \log {\left (3 \right )} + \log {\left (3 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.83, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (3^{x} + 1\right )} \log \left (3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.48, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{\ln \left (3\right )\,\left (3^x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________