Optimal. Leaf size=40 \[ -\frac{\log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{1}{d \log (f) \left (f^{c+d x}+1\right )}+x \]
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Rubi [A] time = 0.0265826, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2282, 44} \[ -\frac{\log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{1}{d \log (f) \left (f^{c+d x}+1\right )}+x \]
Antiderivative was successfully verified.
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Rule 2282
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{1+2 f^{c+d x}+f^{2 c+2 d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (1+x)^2} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{1}{x}-\frac{1}{(1+x)^2}\right ) \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=x+\frac{1}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac{\log \left (1+f^{c+d x}\right )}{d \log (f)}\\ \end{align*}
Mathematica [A] time = 0.0338202, size = 37, normalized size = 0.92 \[ \frac{\frac{1}{f^{c+d x}+1}-\log \left (f^{c+d x}+1\right )+d x \log (f)}{d \log (f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 68, normalized size = 1.7 \begin{align*}{\frac{1}{{{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}+1} \left ( x+x{{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}-{\frac{{{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}}{d\ln \left ( f \right ) }} \right ) }-{\frac{\ln \left ({{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}+1 \right ) }{d\ln \left ( f \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989898, size = 74, normalized size = 1.85 \begin{align*} -\frac{\log \left (f^{d x + c} + 1\right )}{d \log \left (f\right )} + \frac{\log \left (f^{d x + c}\right )}{d \log \left (f\right )} + \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 = 1.60264, size = 159, normalized size = 3.98 \begin{align*} \frac{d f^{d x + c} x \log \left (f\right ) + d x \log \left (f\right ) -{\left (f^{d x + c} + 1\right )} \log \left (f^{d x + c} + 1\right ) + 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 = 0.116324, size = 34, normalized size = 0.85 \begin{align*} x + \frac{1}{d f^{c + d x} \log{\left (f \right )} + d \log{\left (f \right )}} - \frac{\log{\left (f^{c + d x} + 1 \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^{2 \, d x + 2 \, c} + 2 \, f^{d x + c} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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