3.534 \(\int \frac{x^2}{2+f^{-c-d x}+f^{c+d x}} \, dx\)

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]

x^2/(d*Log[f]) - x^2/(d*(1 + f^(c + d*x))*Log[f]) - (2*x*Log[1 + f^(c + d*x)])/(
d^2*Log[f]^2) - (2*PolyLog[2, -f^(c + d*x)])/(d^3*Log[f]^3)

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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]

x^2/(d*Log[f]) - x^2/(d*(1 + f^(c + d*x))*Log[f]) - (2*x*Log[1 + f^(c + d*x)])/(
d^2*Log[f]^2) - (2*PolyLog[2, -f^(c + d*x)])/(d^3*Log[f]^3)

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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]

Timed 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]

(d*x*Log[f]*((d*f^(c + d*x)*x*Log[f])/(1 + f^(c + d*x)) - 2*Log[1 + f^(c + d*x)]
) - 2*PolyLog[2, -f^(c + d*x)])/(d^3*Log[f]^3)

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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]

1/ln(f)/d*x^2/(f^(-d*x-c)+1)-x^2/d/ln(f)-2/d^2/ln(f)*c*x-1/d^3/ln(f)*c^2-2/d^2/l
n(f)^2*ln(f^(-d*x-c)+1)*x+2/d^3/ln(f)^3*polylog(2,-f^(-d*x-c))-2/d^3/ln(f)^2*c*l
n(f^(-d*x-c))

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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]

-x^2/(d*f^(d*x)*f^c*log(f) + d*log(f)) + log(f^(d*x))^2/(d^3*log(f)^3) - 2*(log(
f^(d*x)*f^c + 1)*log(f^(d*x)) + dilog(-f^(d*x)*f^c))/(d^3*log(f)^3)

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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]

-(c^2*log(f)^2 - (d^2*x^2 - c^2)*f^(d*x + c)*log(f)^2 + 2*(f^(d*x + c) + 1)*dilo
g(-f^(d*x + c)) + 2*(d*f^(d*x + c)*x*log(f) + d*x*log(f))*log(f^(d*x + c) + 1))/
(d^3*f^(d*x + c)*log(f)^3 + d^3*log(f)^3)

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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]

-x**2/(d*f**(c + d*x)*log(f) + d*log(f)) + 2*Integral(x/(exp(c*log(f))*exp(d*x*l
og(f)) + 1), x)/(d*log(f))

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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]

integrate(x^2/(f^(d*x + c) + f^(-d*x - c) + 2), x)