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 = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 44
Rule 2282
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.43, size = 59, normalized size = 1.48 \[ \frac {d f^{d x + c} x \log \relax (f) + d x \log \relax (f) - {\left (f^{d x + c} + 1\right )} \log \left (f^{d x + c} + 1\right ) + 1}{d f^{d x + c} \log \relax (f) + d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 68, normalized size = 1.70 \[ -\frac {\ln \left ({\mathrm e}^{\left (d x +c \right ) \ln \relax (f )}+1\right )}{d \ln \relax (f )}+\frac {x \,{\mathrm e}^{\left (d x +c \right ) \ln \relax (f )}+x -\frac {{\mathrm e}^{\left (d x +c \right ) \ln \relax (f )}}{d \ln \relax (f )}}{{\mathrm e}^{\left (d x +c \right ) \ln \relax (f )}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 48, normalized size = 1.20 \[ \frac {d x + c}{d} - \frac {\log \left (f^{d x + c} + 1\right )}{d \log \relax (f)} + \frac {1}{d {\left (f^{d x + c} + 1\right )} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 50, normalized size = 1.25 \[ \frac {1}{d\,\ln \relax (f)\,\left (f^{d\,x}\,f^c+1\right )}-\frac {\ln \left (f^{d\,x}\,f^c+1\right )-d\,x\,\ln \relax (f)}{d\,\ln \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 34, normalized size = 0.85 \[ x + \frac {1}{d f^{c + d x} \log {\relax (f )} + d \log {\relax (f )}} - \frac {\log {\left (f^{c + d x} + 1 \right )}}{d \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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