Optimal. Leaf size=75 \[ \frac {x^2}{d \log (f)}-\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)} \]
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Rubi [A]
time = 0.33, 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 = {2299, 6820,
2222, 2215, 2221, 2317, 2438} \begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2222
Rule 2299
Rule 2317
Rule 2438
Rule 6820
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 \text {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 \begin {gather*} \frac {d x \log (f) \left (\frac {d f^{c+d x} x \log (f)}{1+f^{c+d x}}-2 \log \left (1+f^{c+d x}\right )\right )-2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 134, normalized size = 1.79
method | result | size |
risch | \(\frac {x^{2}}{d \ln \left (f \right ) \left (f^{-d x -c}+1\right )}-\frac {x^{2}}{d \ln \left (f \right )}-\frac {2 c x}{d^{2} \ln \left (f \right )}-\frac {c^{2}}{d^{3} \ln \left (f \right )}-\frac {2 \ln \left (1+f^{-d x} f^{-c}\right ) x}{\ln \left (f \right )^{2} d^{2}}+\frac {2 \polylog \left (2, -f^{-d x} f^{-c}\right )}{\ln \left (f \right )^{3} d^{3}}-\frac {2 c \ln \left (f^{-d x} f^{-c}\right )}{\ln \left (f \right )^{2} d^{3}}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 74, normalized size = 0.99 \begin {gather*} -\frac {x^{2}}{d f^{d x} f^{c} \log \left (f\right ) + d \log \left (f\right )} + \frac {x^{2}}{d \log \left (f\right )} - \frac {2 \, {\left (d x \log \left (f^{d x} f^{c} + 1\right ) \log \left (f\right ) + {\rm Li}_2\left (-f^{d x} f^{c}\right )\right )}}{d^{3} \log \left (f\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 114, normalized size = 1.52 \begin {gather*} -\frac {c^{2} \log \left (f\right )^{2} - {\left (d^{2} x^{2} - c^{2}\right )} f^{d x + c} \log \left (f\right )^{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 \left (f\right ) + d x \log \left (f\right )\right )} \log \left (f^{d x + c} + 1\right )}{d^{3} f^{d x + c} \log \left (f\right )^{3} + d^{3} \log \left (f\right )^{3}} \end {gather*}
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 \begin {gather*} - \frac {x^{2}}{d f^{c + d x} \log {\left (f \right )} + d \log {\left (f \right )}} + \frac {2 \int \frac {x}{e^{c \log {\left (f \right )}} e^{d x \log {\left (f \right )}} + 1}\, dx}{d \log {\left (f \right )}} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{\frac {1}{f^{c+d\,x}}+f^{c+d\,x}+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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