Optimal. Leaf size=20 \[ -\frac {1}{d \log (f) \left (f^{c+d x}+1\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2282, 32} \[ -\frac {1}{d \log (f) \left (f^{c+d x}+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{2+f^{-c-d x}+f^{c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=-\frac {1}{d \left (1+f^{c+d x}\right ) \log (f)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \[ -\frac {1}{d \log (f) \left (f^{c+d x}+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 20, normalized size = 1.00 \[ -\frac {1}{d f^{d x + c} \log \relax (f) + d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 22, normalized size = 1.10 \[ -\frac {1}{{\left (f^{d x} f^{c} + 1\right )} d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 25, normalized size = 1.25 \[ \frac {1}{\left ({\mathrm e}^{\left (-d x -c \right ) \ln \relax (f )}+1\right ) d \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.93, size = 22, normalized size = 1.10 \[ \frac {1}{d {\left (f^{-d x - c} + 1\right )} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.56, size = 20, normalized size = 1.00 \[ -\frac {1}{d\,\ln \relax (f)\,\left (f^{c+d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.11, size = 19, normalized size = 0.95 \[ - \frac {1}{d f^{c + d x} \log {\relax (f )} + d \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________