Optimal. Leaf size=145 \[ \frac {2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}+\frac {2 \text {Li}_3\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {2 x \text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 x \log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x^2 \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac {x^2}{d \log (f) \left (f^{c+d x}+1\right )}-\frac {x^2}{d \log (f)}+\frac {x^3}{3} \]
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Rubi [A] time = 0.42, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6688, 2185, 2184, 2190, 2531, 2282, 6589, 2191, 2279, 2391} \[ -\frac {2 x \text {PolyLog}\left (2,-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 \text {PolyLog}\left (2,-f^{c+d x}\right )}{d^3 \log ^3(f)}+\frac {2 \text {PolyLog}\left (3,-f^{c+d x}\right )}{d^3 \log ^3(f)}+\frac {2 x \log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x^2 \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac {x^2}{d \log (f) \left (f^{c+d x}+1\right )}-\frac {x^2}{d \log (f)}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 6589
Rule 6688
Rubi steps
\begin {align*} \int \frac {x^2}{1+2 f^{c+d x}+f^{2 c+2 d x}} \, dx &=\int \frac {x^2}{\left (1+f^{c+d x}\right )^2} \, dx\\ &=-\int \frac {f^{c+d x} x^2}{\left (1+f^{c+d x}\right )^2} \, dx+\int \frac {x^2}{1+f^{c+d x}} \, dx\\ &=\frac {x^3}{3}+\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)}-\int \frac {f^{c+d x} x^2}{1+f^{c+d x}} \, dx\\ &=\frac {x^3}{3}-\frac {x^2}{d \log (f)}+\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}+\frac {2 \int \frac {f^{c+d x} x}{1+f^{c+d x}} \, dx}{d \log (f)}+\frac {2 \int x \log \left (1+f^{c+d x}\right ) \, dx}{d \log (f)}\\ &=\frac {x^3}{3}-\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 {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac {2 x \text {Li}_2\left (-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 {2 \int \text {Li}_2\left (-f^{c+d x}\right ) \, dx}{d^2 \log ^2(f)}\\ &=\frac {x^3}{3}-\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 {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac {2 x \text {Li}_2\left (-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 {2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}\\ &=\frac {x^3}{3}-\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 {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}+\frac {2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {2 x \text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 \text {Li}_3\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 123, normalized size = 0.85 \[ \frac {-\frac {3 d^2 x^2 \log ^2(f) \left (f^{c+d x}+\left (f^{c+d x}+1\right ) \log \left (f^{c+d x}+1\right )\right )}{f^{c+d x}+1}+6 \text {Li}_3\left (-f^{c+d x}\right )+(6-6 d x \log (f)) \text {Li}_2\left (-f^{c+d x}\right )+6 d x \log (f) \log \left (f^{c+d x}+1\right )+d^3 x^3 \log ^3(f)}{3 d^3 \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.43, size = 210, normalized size = 1.45 \[ \frac {3 \, c^{2} \log \relax (f)^{2} + {\left (d^{3} x^{3} + c^{3}\right )} \log \relax (f)^{3} + {\left ({\left (d^{3} x^{3} + c^{3}\right )} \log \relax (f)^{3} - 3 \, {\left (d^{2} x^{2} - c^{2}\right )} \log \relax (f)^{2}\right )} f^{d x + c} - 6 \, {\left (d x \log \relax (f) + {\left (d x \log \relax (f) - 1\right )} f^{d x + c} - 1\right )} {\rm Li}_2\left (-f^{d x + c}\right ) - 3 \, {\left (d^{2} x^{2} \log \relax (f)^{2} - 2 \, d x \log \relax (f) + {\left (d^{2} x^{2} \log \relax (f)^{2} - 2 \, d x \log \relax (f)\right )} f^{d x + c}\right )} \log \left (f^{d x + c} + 1\right ) + 6 \, {\left (f^{d x + c} + 1\right )} {\rm polylog}\left (3, -f^{d x + c}\right )}{3 \, {\left (d^{3} f^{d x + c} \log \relax (f)^{3} + d^{3} \log \relax (f)^{3}\right )}} \]
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 \[ \int \frac {x^{2}}{f^{2 \, d x + 2 \, c} + 2 \, f^{d x + c} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 232, normalized size = 1.60 \[ \frac {x^{3}}{3}-\frac {c^{2} x}{d^{2}}-\frac {x^{2} \ln \left (f^{c} f^{d x}+1\right )}{d \ln \relax (f )}-\frac {2 c^{3}}{3 d^{3}}+\frac {x^{2}}{\left (f^{d x +c}+1\right ) d \ln \relax (f )}-\frac {x^{2}}{d \ln \relax (f )}+\frac {c^{2} \ln \left (f^{c} f^{d x}\right )}{d^{3} \ln \relax (f )}-\frac {2 c x}{d^{2} \ln \relax (f )}-\frac {c^{2}}{d^{3} \ln \relax (f )}-\frac {2 x \polylog \left (2, -f^{c} f^{d x}\right )}{d^{2} \ln \relax (f )^{2}}+\frac {2 x \ln \left (f^{c} f^{d x}+1\right )}{d^{2} \ln \relax (f )^{2}}+\frac {2 c \ln \left (f^{c} f^{d x}\right )}{d^{3} \ln \relax (f )^{2}}+\frac {2 \polylog \left (2, -f^{c} f^{d x}\right )}{d^{3} \ln \relax (f )^{3}}+\frac {2 \polylog \left (3, -f^{c} f^{d x}\right )}{d^{3} \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 159, normalized size = 1.10 \[ \frac {x^{2}}{d f^{d x} f^{c} \log \relax (f) + d \log \relax (f)} + \frac {d^{3} x^{3} \log \relax (f)^{3} - 3 \, d^{2} x^{2} \log \relax (f)^{2}}{3 \, d^{3} \log \relax (f)^{3}} - \frac {d^{2} x^{2} \log \left (f^{d x} f^{c} + 1\right ) \log \relax (f)^{2} + 2 \, d x {\rm Li}_2\left (-f^{d x} f^{c}\right ) \log \relax (f) - 2 \, {\rm Li}_{3}(-f^{d x} f^{c})}{d^{3} \log \relax (f)^{3}} + \frac {2 \, {\left (d x \log \left (f^{d x} f^{c} + 1\right ) \log \relax (f) + {\rm Li}_2\left (-f^{d x} f^{c}\right )\right )}}{d^{3} \log \relax (f)^{3}} \]
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 \[ \int \frac {x^2}{f^{2\,c+2\,d\,x}+2\,f^{c+d\,x}+1} \,d x \]
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 \[ \frac {x^{2}}{d f^{c + d x} \log {\relax (f )} + d \log {\relax (f )}} + \frac {\int \left (- \frac {2 x}{e^{c \log {\relax (f )}} e^{d x \log {\relax (f )}} + 1}\right )\, dx + \int \frac {d x^{2} \log {\relax (f )}}{e^{c \log {\relax (f )}} e^{d x \log {\relax (f )}} + 1}\, dx}{d \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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