Optimal. Leaf size=75 \[ -\frac {2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {2 x \log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x^2}{d \log (f) \left (f^{c+d x}+1\right )}+\frac {x^2}{d \log (f)} \]
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Rubi [A] time = 0.49, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2267, 6688, 2191, 2184, 2190, 2279, 2391} \[ -\frac {2 \text {PolyLog}\left (2,-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {2 x \log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x^2}{d \log (f) \left (f^{c+d x}+1\right )}+\frac {x^2}{d \log (f)} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2191
Rule 2267
Rule 2279
Rule 2391
Rule 6688
Rubi steps
\begin {align*} \int \frac {x^2}{2+f^{-c-d x}+f^{c+d x}} \, dx &=\int \frac {f^{c+d x} x^2}{1+2 f^{c+d x}+f^{2 (c+d x)}} \, dx\\ &=\int \frac {f^{c+d x} x^2}{\left (1+f^{c+d x}\right )^2} \, dx\\ &=-\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {2 \int \frac {x}{1+f^{c+d x}} \, dx}{d \log (f)}\\ &=\frac {x^2}{d \log (f)}-\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {2 \int \frac {f^{c+d x} x}{1+f^{c+d x}} \, dx}{d \log (f)}\\ &=\frac {x^2}{d \log (f)}-\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 \int \log \left (1+f^{c+d x}\right ) \, dx}{d^2 \log ^2(f)}\\ &=\frac {x^2}{d \log (f)}-\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}\\ &=\frac {x^2}{d \log (f)}-\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 63, normalized size = 0.84 \[ \frac {d x \log (f) \left (\frac {d x \log (f) f^{c+d x}}{f^{c+d x}+1}-2 \log \left (f^{c+d x}+1\right )\right )-2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 114, normalized size = 1.52 \[ -\frac {c^{2} \log \relax (f)^{2} - {\left (d^{2} x^{2} - c^{2}\right )} f^{d x + c} \log \relax (f)^{2} + 2 \, {\left (f^{d x + c} + 1\right )} {\rm Li}_2\left (-f^{d x + c}\right ) + 2 \, {\left (d f^{d x + c} x \log \relax (f) + d x \log \relax (f)\right )} \log \left (f^{d x + c} + 1\right )}{d^{3} f^{d x + c} \log \relax (f)^{3} + d^{3} \log \relax (f)^{3}} \]
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 \[ \int \frac {x^{2}}{f^{d x + c} + f^{-d x - c} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 134, normalized size = 1.79 \[ \frac {x^{2}}{\left (f^{-d x -c}+1\right ) d \ln \relax (f )}-\frac {x^{2}}{d \ln \relax (f )}-\frac {2 c x}{d^{2} \ln \relax (f )}-\frac {c^{2}}{d^{3} \ln \relax (f )}-\frac {2 x \ln \left (f^{-c} f^{-d x}+1\right )}{d^{2} \ln \relax (f )^{2}}-\frac {2 c \ln \left (f^{-c} f^{-d x}\right )}{d^{3} \ln \relax (f )^{2}}+\frac {2 \polylog \left (2, -f^{-c} f^{-d x}\right )}{d^{3} \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 74, normalized size = 0.99 \[ -\frac {x^{2}}{d f^{d x} f^{c} \log \relax (f) + d \log \relax (f)} + \frac {x^{2}}{d \log \relax (f)} - \frac {2 \, {\left (d x \log \left (f^{d x} f^{c} + 1\right ) \log \relax (f) + {\rm Li}_2\left (-f^{d x} f^{c}\right )\right )}}{d^{3} \log \relax (f)^{3}} \]
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 \[ \int \frac {x^2}{\frac {1}{f^{c+d\,x}}+f^{c+d\,x}+2} \,d x \]
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 \[ - \frac {x^{2}}{d f^{c + d x} \log {\relax (f )} + d \log {\relax (f )}} + \frac {2 \int \frac {x}{e^{c \log {\relax (f )}} e^{d x \log {\relax (f )}} + 1}\, dx}{d \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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