Optimal. Leaf size=50 \[ \frac {x}{d \log (f)}-\frac {x}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {\log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)} \]
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Rubi [A]
time = 0.20, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2299, 6820,
2222, 2320, 36, 29, 31} \begin {gather*} -\frac {\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x}{d \log (f) \left (f^{c+d x}+1\right )}+\frac {x}{d \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2222
Rule 2299
Rule 2320
Rule 6820
Rubi steps
\begin {align*} \int \frac {x}{2+f^{-c-d x}+f^{c+d x}} \, dx &=\int \frac {f^{c+d x} x}{1+2 f^{c+d x}+f^{2 (c+d x)}} \, dx\\ &=\int \frac {f^{c+d x} x}{\left (1+f^{c+d x}\right )^2} \, dx\\ &=-\frac {x}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {\int \frac {1}{1+f^{c+d x}} \, dx}{d \log (f)}\\ &=-\frac {x}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {\text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}\\ &=-\frac {x}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}\\ &=\frac {x}{d \log (f)}-\frac {x}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {\log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 44, normalized size = 0.88 \begin {gather*} \frac {\frac {d f^{c+d x} x \log (f)}{1+f^{c+d x}}-\log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 64, normalized size = 1.28
method | result | size |
norman | \(-\frac {x \,{\mathrm e}^{\left (-d x -c \right ) \ln \left (f \right )}}{d \ln \left (f \right ) \left ({\mathrm e}^{\left (-d x -c \right ) \ln \left (f \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{\left (-d x -c \right ) \ln \left (f \right )}+1\right )}{d^{2} \ln \left (f \right )^{2}}\) | \(64\) |
risch | \(-\frac {x}{d \ln \left (f \right )}-\frac {c}{d^{2} \ln \left (f \right )}+\frac {x}{d \ln \left (f \right ) \left (f^{-d x -c}+1\right )}-\frac {\ln \left (f^{-d x -c}+1\right )}{d^{2} \ln \left (f \right )^{2}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 57, normalized size = 1.14 \begin {gather*} \frac {f^{d x} f^{c} x}{d f^{d x} f^{c} \log \left (f\right ) + d \log \left (f\right )} - \frac {\log \left (\frac {f^{d x} f^{c} + 1}{f^{c}}\right )}{d^{2} \log \left (f\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 61, normalized size = 1.22 \begin {gather*} \frac {d f^{d x + c} x \log \left (f\right ) - {\left (f^{d x + c} + 1\right )} \log \left (f^{d x + c} + 1\right )}{d^{2} f^{d x + c} \log \left (f\right )^{2} + d^{2} \log \left (f\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 42, normalized size = 0.84 \begin {gather*} - \frac {x}{d f^{c + d x} \log {\left (f \right )} + d \log {\left (f \right )}} + \frac {x}{d \log {\left (f \right )}} - \frac {\log {\left (f^{c + d x} + 1 \right )}}{d^{2} \log {\left (f \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.62, size = 52, normalized size = 1.04 \begin {gather*} \frac {f^{d\,x}\,f^c\,x}{d\,\ln \left (f\right )\,\left (f^{d\,x}\,f^c+1\right )}-\frac {\ln \left (f^{d\,x}\,f^c+1\right )}{d^2\,{\ln \left (f\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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