Optimal. Leaf size=50 \[ -\frac {\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x}{d \log (f) \left (f^{c+d x}+1\right )}+\frac {x}{d \log (f)} \]
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Rubi [A] time = 0.29, 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 = {2267, 6688, 2191, 2282, 36, 29, 31} \[ -\frac {\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x}{d \log (f) \left (f^{c+d x}+1\right )}+\frac {x}{d \log (f)} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2191
Rule 2267
Rule 2282
Rule 6688
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 {\operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {\operatorname {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 = 0.07, size = 44, normalized size = 0.88 \[ \frac {\frac {d x \log (f) f^{c+d x}}{f^{c+d x}+1}-\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 61, normalized size = 1.22 \[ \frac {d f^{d x + c} x \log \relax (f) - {\left (f^{d x + c} + 1\right )} \log \left (f^{d x + c} + 1\right )}{d^{2} f^{d x + c} \log \relax (f)^{2} + d^{2} \log \relax (f)^{2}} \]
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 \[ \int \frac {x}{f^{d x + c} + f^{-d x - c} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 64, normalized size = 1.28 \[ -\frac {x \,{\mathrm e}^{\left (-d x -c \right ) \ln \relax (f )}}{\left ({\mathrm e}^{\left (-d x -c \right ) \ln \relax (f )}+1\right ) d \ln \relax (f )}-\frac {\ln \left ({\mathrm e}^{\left (-d x -c \right ) \ln \relax (f )}+1\right )}{d^{2} \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 57, normalized size = 1.14 \[ \frac {f^{d x} f^{c} x}{d f^{d x} f^{c} \log \relax (f) + d \log \relax (f)} - \frac {\log \left (\frac {f^{d x} f^{c} + 1}{f^{c}}\right )}{d^{2} \log \relax (f)^{2}} \]
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 \[ \frac {f^{d\,x}\,f^c\,x}{d\,\ln \relax (f)\,\left (f^{d\,x}\,f^c+1\right )}-\frac {\ln \left (f^{d\,x}\,f^c+1\right )}{d^2\,{\ln \relax (f)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 42, normalized size = 0.84 \[ - \frac {x}{d f^{c + d x} \log {\relax (f )} + d \log {\relax (f )}} + \frac {x}{d \log {\relax (f )}} - \frac {\log {\left (f^{c + d x} + 1 \right )}}{d^{2} \log {\relax (f )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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