Optimal. Leaf size=20 \[ -\frac{1}{d \log (f) \left (f^{c+d x}+1\right )} \]
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Rubi [A] time = 0.0203003, antiderivative size = 20, normalized size of antiderivative = 1., 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.
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Rule 2282
Rule 32
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 = 0.0172876, size = 20, normalized size = 1. \[ -\frac{1}{d \log (f) \left (f^{c+d x}+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 25, normalized size = 1.3 \begin{align*}{\frac{1}{d\ln \left ( f \right ) \left ({{\rm e}^{ \left ( -dx-c \right ) \ln \left ( f \right ) }}+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953765, size = 30, normalized size = 1.5 \begin{align*} \frac{1}{d{\left (f^{-d x - c} + 1\right )} \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25229, size = 51, normalized size = 2.55 \begin{align*} -\frac{1}{d f^{d x + c} \log \left (f\right ) + d \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.102194, size = 19, normalized size = 0.95 \begin{align*} - \frac{1}{d f^{c + d x} \log{\left (f \right )} + d \log{\left (f \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{f^{d x + c} + f^{-d x - c} + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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