Optimal. Leaf size=40 \[ x+\frac {1}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {\log \left (1+f^{c+d x}\right )}{d \log (f)} \]
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Rubi [A]
time = 0.02, 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 = {2320, 46}
\begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{1+2 f^{c+d x}+f^{2 c+2 d x}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x (1+x)^2} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=\frac {\text {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 = 0.06, size = 44, normalized size = 1.10 \begin {gather*} \frac {\frac {1}{1+f^{c+d x}}+\log \left (f^{c+d x}\right )-\log \left (d \left (1+f^{c+d x}\right ) \log (f)\right )}{d \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 46, normalized size = 1.15
method | result | size |
risch | \(x +\frac {c}{d}+\frac {1}{d \left (1+f^{d x +c}\right ) \ln \left (f \right )}-\frac {\ln \left (1+f^{d x +c}\right )}{d \ln \left (f \right )}\) | \(46\) |
norman | \(\frac {x +x \,{\mathrm e}^{\left (d x +c \right ) \ln \left (f \right )}+\frac {1}{d \ln \left (f \right )}}{{\mathrm e}^{\left (d x +c \right ) \ln \left (f \right )}+1}-\frac {\ln \left ({\mathrm e}^{\left (d x +c \right ) \ln \left (f \right )}+1\right )}{d \ln \left (f \right )}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 48, normalized size = 1.20 \begin {gather*} \frac {d x + c}{d} - \frac {\log \left (f^{d x + c} + 1\right )}{d \log \left (f\right )} + \frac {1}{d {\left (f^{d x + c} + 1\right )} \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 59, normalized size = 1.48 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 34, normalized size = 0.85 \begin {gather*} 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 {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
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 \begin {gather*} \frac {1}{d\,\ln \left (f\right )\,\left (f^{d\,x}\,f^c+1\right )}-\frac {\ln \left (f^{d\,x}\,f^c+1\right )-d\,x\,\ln \left (f\right )}{d\,\ln \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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