Optimal. Leaf size=96 \[ -\frac {\text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac {x}{d \log (f) \left (f^{c+d x}+1\right )}-\frac {x}{d \log (f)}+\frac {x^2}{2} \]
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Rubi [A] time = 0.27, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {6688, 2185, 2184, 2190, 2279, 2391, 2191, 2282, 36, 29, 31} \[ -\frac {\text {PolyLog}\left (2,-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac {x}{d \log (f) \left (f^{c+d x}+1\right )}-\frac {x}{d \log (f)}+\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 6688
Rubi steps
\begin {align*} \int \frac {x}{1+2 f^{c+d x}+f^{2 c+2 d x}} \, dx &=\int \frac {x}{\left (1+f^{c+d x}\right )^2} \, dx\\ &=-\int \frac {f^{c+d x} x}{\left (1+f^{c+d x}\right )^2} \, dx+\int \frac {x}{1+f^{c+d x}} \, dx\\ &=\frac {x^2}{2}+\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)}-\int \frac {f^{c+d x} x}{1+f^{c+d x}} \, dx\\ &=\frac {x^2}{2}+\frac {x}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {x \log \left (1+f^{c+d x}\right )}{d \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 {\int \log \left (1+f^{c+d x}\right ) \, dx}{d \log (f)}\\ &=\frac {x^2}{2}+\frac {x}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {x \log \left (1+f^{c+d x}\right )}{d \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 {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}\\ &=\frac {x^2}{2}-\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)}-\frac {x \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac {\text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 88, normalized size = 0.92 \[ -\frac {\text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}+\frac {1}{2} x \left (\frac {2}{d \log (f) f^{c+d x}+d \log (f)}+x\right )-\frac {x \left (\log \left (f^{c+d x}+1\right )+1\right )}{d \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 143, normalized size = 1.49 \[ \frac {{\left (d^{2} x^{2} - c^{2}\right )} \log \relax (f)^{2} + {\left ({\left (d^{2} x^{2} - c^{2}\right )} \log \relax (f)^{2} - 2 \, {\left (d x + c\right )} \log \relax (f)\right )} f^{d x + c} - 2 \, {\left (f^{d x + c} + 1\right )} {\rm Li}_2\left (-f^{d x + c}\right ) - 2 \, {\left (d x \log \relax (f) + {\left (d x \log \relax (f) - 1\right )} f^{d x + c} - 1\right )} \log \left (f^{d x + c} + 1\right ) - 2 \, c \log \relax (f)}{2 \, {\left (d^{2} f^{d x + c} \log \relax (f)^{2} + d^{2} \log \relax (f)^{2}\right )}} \]
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 \[ \int \frac {x}{f^{2 \, d x + 2 \, c} + 2 \, f^{d x + c} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 143, normalized size = 1.49 \[ \frac {x^{2}}{2}+\frac {c x}{d}+\frac {c^{2}}{2 d^{2}}-\frac {x \ln \left (f^{c} f^{d x}+1\right )}{d \ln \relax (f )}-\frac {c \ln \left (f^{c} f^{d x}\right )}{d^{2} \ln \relax (f )}+\frac {x}{\left (f^{d x +c}+1\right ) d \ln \relax (f )}-\frac {\polylog \left (2, -f^{c} f^{d x}\right )}{d^{2} \ln \relax (f )^{2}}-\frac {\ln \left (f^{c} f^{d x}\right )}{d^{2} \ln \relax (f )^{2}}+\frac {\ln \left (f^{c} f^{d x}+1\right )}{d^{2} \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 95, normalized size = 0.99 \[ \frac {1}{2} \, x^{2} + \frac {x}{d f^{d x} f^{c} \log \relax (f) + d \log \relax (f)} - \frac {x}{d \log \relax (f)} - \frac {d x \log \left (f^{d x} f^{c} + 1\right ) \log \relax (f) + {\rm Li}_2\left (-f^{d x} f^{c}\right )}{d^{2} \log \relax (f)^{2}} + \frac {\log \left (f^{d x} f^{c} + 1\right )}{d^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{f^{2\,c+2\,d\,x}+2\,f^{c+d\,x}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x}{d f^{c + d x} \log {\relax (f )} + d \log {\relax (f )}} + \frac {\int \frac {d x \log {\relax (f )}}{e^{c \log {\relax (f )}} e^{d x \log {\relax (f )}} + 1}\, dx + \int \left (- \frac {1}{e^{c \log {\relax (f )}} e^{d x \log {\relax (f )}} + 1}\right )\, dx}{d \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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