Optimal. Leaf size=75 \[ -\frac{2 \text{PolyLog}\left (2,-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac{2 x \log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac{x^2}{d \log (f) \left (f^{c+d x}+1\right )}+\frac{x^2}{d \log (f)} \]
[Out]
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Rubi [A] time = 0.754997, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{2 \text{PolyLog}\left (2,-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac{2 x \log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac{x^2}{d \log (f) \left (f^{c+d x}+1\right )}+\frac{x^2}{d \log (f)} \]
Antiderivative was successfully verified.
[In] Int[x^2/(2 + f^(-c - d*x) + f^(c + d*x)),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(2+f**(-d*x-c)+f**(d*x+c)),x)
[Out]
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Mathematica [A] time = 0.146475, size = 63, normalized size = 0.84 \[ \frac{d x \log (f) \left (\frac{d x \log (f) f^{c+d x}}{f^{c+d x}+1}-2 \log \left (f^{c+d x}+1\right )\right )-2 \text{PolyLog}\left (2,-f^{c+d x}\right )}{d^3 \log ^3(f)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(2 + f^(-c - d*x) + f^(c + d*x)),x]
[Out]
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Maple [A] time = 0.043, size = 129, normalized size = 1.7 \[{\frac{{x}^{2}}{d\ln \left ( f \right ) \left ({f}^{-dx-c}+1 \right ) }}-{\frac{{x}^{2}}{d\ln \left ( f \right ) }}-2\,{\frac{cx}{{d}^{2}\ln \left ( f \right ) }}-{\frac{{c}^{2}}{{d}^{3}\ln \left ( f \right ) }}-2\,{\frac{\ln \left ({f}^{-dx-c}+1 \right ) x}{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}}+2\,{\frac{{\it polylog} \left ( 2,-{f}^{-dx-c} \right ) }{{d}^{3} \left ( \ln \left ( f \right ) \right ) ^{3}}}-2\,{\frac{c\ln \left ({f}^{-dx-c} \right ) }{{d}^{3} \left ( \ln \left ( f \right ) \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(2+f^(-d*x-c)+f^(d*x+c)),x)
[Out]
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Maxima [A] time = 0.888333, size = 109, normalized size = 1.45 \[ -\frac{x^{2}}{d f^{d x} f^{c} \log \left (f\right ) + d \log \left (f\right )} + \frac{\log \left (f^{d x}\right )^{2}}{d^{3} \log \left (f\right )^{3}} - \frac{2 \,{\left (\log \left (f^{d x} f^{c} + 1\right ) \log \left (f^{d x}\right ) +{\rm Li}_2\left (-f^{d x} f^{c}\right )\right )}}{d^{3} \log \left (f\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(f^(d*x + c) + f^(-d*x - c) + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253836, size = 154, normalized size = 2.05 \[ -\frac{c^{2} \log \left (f\right )^{2} -{\left (d^{2} x^{2} - c^{2}\right )} f^{d x + c} \log \left (f\right )^{2} + 2 \,{\left (f^{d x + c} + 1\right )}{\rm Li}_2\left (-f^{d x + c}\right ) + 2 \,{\left (d f^{d x + c} x \log \left (f\right ) + d x \log \left (f\right )\right )} \log \left (f^{d x + c} + 1\right )}{d^{3} f^{d x + c} \log \left (f\right )^{3} + d^{3} \log \left (f\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(f^(d*x + c) + f^(-d*x - c) + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{x^{2}}{d f^{c + d x} \log{\left (f \right )} + d \log{\left (f \right )}} + \frac{2 \int \frac{x}{e^{c \log{\left (f \right )}} e^{d x \log{\left (f \right )}} + 1}\, dx}{d \log{\left (f \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(2+f**(-d*x-c)+f**(d*x+c)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{f^{d x + c} + f^{-d x - c} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(f^(d*x + c) + f^(-d*x - c) + 2),x, algorithm="giac")
[Out]